| Where Recherche duTemps Perdu
---- meets Kirchliche Dogmatik
Alaska 3: Some Fauna
[I really wanted to do a StreetJelly set this afternoon, but still felt a bit too crummy to make it work.]
I'm going to use a few pictures from my 2008 study trip on this entry, and I'll mark them as such.
Please take it from me: There are no moose on Barinoff Island, where Sitka is located.
However, there are bears.
And now you know the reason why there are no moose.
That whole area of the southern panhandle of Alaska constitutes rain forest. Now, some of you may feel confused because that expression has frequently been used as synonymous with "jungle." Jungles are tropical rain forests, but there are also temperate rain forests, e.g. the Alaskan panhandle. See the Wikipedia article on "Rain Forest," which only briefly acknowledges the existence of temperate rain forests.
Did I just write "temperate" with regard to Alaskan weather? I did, and I did so deliberately. There are warm waters off the shores of British Columbia and Alaska, streaming in a current called he "Gulf of Japan." Just to illustrate, here are the weather predictions for the next ten days, including today (Sunday) for Smalltown, USA and Sitka, Alaska. I got these numbers from the Microsoft Weather app, and I'm not providing any guarantee for accuracy. But even if the exact numbers all turn out differently, you'll still have the same pattern.
Well, it turns out that Bravenet doesn't want to show my table in the manner in which I created it. I will try to construct along some other method. In the mean time, Sitka's temperatures were higher; the differential between night and day was less, and every day showed some precipitation.Â
Of all of the wild life in Sitka, as well as the surrounding area, none are as important as two birds: the raven and the eagle. Please hang in here for a moment as I write this without sufficient explanation. In the belief system of the Tlingit nation (the original settlers) there is a hierarchy in which the raven ranks a little bit higher than the Eagle. The adjoining Haida people reverse this relationship.Looks like I couldn't have picked a more illustrative set of days. Only on two days are the temperatures expected to be higher in Smalltown. For the most part, the Sitka temperatures are higher, and the differential between high and low is far less. And, finally, there is not a day listed for Sitka without precipitation, whereas it just so happens that none is expected here in Hoosierland for the next ten days. In case you're wondering how you can have snow when the temperatures are above freezing, the snow predictions apply to the mountains surrounding Sitka. Some of it will show up in town, but--given the temperatures--will not stick.
Ravens are a constant presence year-round. If you can't see them for a moment, you can certainly hear them.
During the summer, bald-headed eagles are everywhere in and out of town, but in late November they were a little more rare. However, on a walk through the small national park, Wolf and I discovered one of them along with three juveniles. Below is an animation of four pictures.
To Wolf's surprise (not mine since I didn't know anything about it), there still were salmon in the Indian River inlet. I took some video clips, and I must warn you that you may want to take some anti-motion-sickness pill before watching. The salmon gets a little more distinctive about half way through. You hear ravens cawing in the background. At the outset you hear some clucking sounds; those are made by ravens as well.
Finally, here is a picture of the hole my leg went through as I was filming the salmon.
Next time: Living with/in/alongside Nature.
The woods are lovely dark and deep ...
I know. That line comes shortly before the one I used in the last entry, and I'm not being creative. But really: I'm just being economical since it occurred to me that I could mine yet another line from "Stopping by Woods on a Snowy Evening" by Robert Frost, one of my favorites.
In order to get into a manageable rhythm again, I'm thinking of devoting one entry each to what I learned about some plants and animals. After that, we'll make a few observations about the culture.
On the first day there, Wolf and I went for a little walk through a forested area right at the edge of Sitka. He has been a tour guide in Sitka for many decades, even before he retired from the police department. Combined with his third cap as a naturalist, it is impossible to walk with him without his frequently calling your attention to something fascinating by the wayside that you would have missed otherwise.
For example, he directed me to a fern which was obviously in its cold-weather mode. If it had been summer, the leaves in all three examples below would have stood up vertically. This one's blades are arranged in a tight star-shaped cluster.
It is called a "sword fern" and is quite common, particularly in the North American West.
I was no sooner done taking a couple of pictures of the sword fern, when Wolf was on the other side of the trail, calling me over to greet a "deer fern." It, too, is strongly represented in the West, but it is also found in Europe. I'm guessing that it may have been imported to America, though for all that I know, it may have gone in the other direction.
As you can see, deer fern also likes to bunch, but not nearly as tightly or with as much precision as a sword fern.
The third type of fern Wolf showed me was just another few feet away. I would just have noticed that there were a lot of ferns, but I would not have given a thought to the idea that two genuses and three species in all were living there in the same neighborhood.
The licorice fern shares its genus with the sword fern. It loves the soggy, relatively warmish weather in that part of Alaska and far West British Columbia. In fact, to illustrate the conditions, here is a picture of the moss-covered ground in this part of the forest.
I can't help but think of Terry Brooks's forests in the Shanara series looking at this.
After a while, the forest became lighter. There was more air and what the good people of Sitka call "daylight," and the environment changed in response. Between the somewhat sparser trees there were little "systems" of lychens and sphagnum moss, as illustrated below.
Sphagnum moss doesn't seem like much, but if you look closely at it, you can see how the top of an individual plant form what you may want to call a crown or a star, as Wolf is showing it to me.
I hope that you may have found this little description of some aspects of the flora of Baranoff Island worthwhile. Still, you may be wondering whether all of this is going to lead to some insights into theology or apologetics, or whether we're just looking at some interesting facets of life in the pacific north-north-west.
The answer is, of course, "Yes, I'm heading somewhere with this." But maybe you're a newcomer to my newly revived blog and aren't familiar with my methods. First step: learn the subject matter (which in this case is the cultural and environmental setting of Sitka), then uncover it and analyze it. And, by the way, I doubt that anyone can figure out yet where I'm heading. It may start to become a little clearer next time, when we talk about the fauna of Baranoff Island.
Next time: ravens, eagles, herons, and salmon.
...and miles to go before I sleep.
But sleep I will. And I will wake up alongside "Buggy,"
and all the tears and regrets and loneliness will be gone.
Look! God's dwelling is with humanity,
For those blog readers who may be clueless as to what I'm writing about, my dear wife, June, passed away on October 1 of this year. It had turned out that her increasing weakness and incapacity was due to her breast cancer of 10 years ago. It had never returned to her breasts, thereby yielding clean breast biopsies, but had metastasized throughout her skeleton. You can go back to find my Facebook postings over the last year or so if you're interested in more details.
I have not yet sorted out all of my feelings sufficiently to write a helpful, let alone inspiring, account of my thoughts, doubts, and discoveries during that time. Suffice it to say that a part of it was hellish, and that I am incredibly grateful for everyone who was there for me and for June, as well as those people who would have been if it had been a possibility. -- And for the people who reassured me about the relevance of God's presence at that time.
Here is the little video about June's life I played at her memorial service and then posted on Facebook and YouTube.
Also, before I leave the topic, please let me once more express my gratitude to all those wonderful people who provided financial help so as to keep this personal tragedy from being compounded into a financial disaster. For example, the nursing home's rate for room and board (which Medicare paid for only partially) was around $7,000 a month, so you can be sure that every last penny I received went to June's care.
For the time being, at least, I figured that the only way I could get back into doing my blog was simply to start rather than first provide a detailed prolegomena of the last fourteen months. So, I'm going to jump right into the last couple of weeks and talk about my ten days in Alaska.
When June passed away, my older brother Wolf gave me a choice: Either he could come out to the memorial service, or he would be happy to spend the same amount of money for me to come out to see him and his clan in Sitka.
That wasn't much of a choice for me. Obviously, spending ten days in Sitka with him and his wife, as well as other relatives and relations-by-marriage, was highly preferable to having a short encounter along with a bunch of other visitors around the time of the service. So, I began my trip there on November 20 and started the return on December 1. If you're going to try to calculate the times, please take into account that Alaska time is four hours behind Eastern time.
A panorama view (i.e. actually a semi-circle) of Sitka with Mt. Edgecombe on the left.
Towards the right you see the steeple of St. Michael the Archangel Russian Orthodox Church.
Here's another shot, focusing on St. Michael's through a bit of forested area.
So, presumably you can tell that Sitka, located on Baranoff Island just off the southern end of the Alaskan panhandle, is a wonderful place aesthetically. In case you're wondering about the weather, it pretty much hung around the upper thirties and lower forties (above freezing) the entire time, and it doesn't get as cold there as it does here in Indiana.
The trip out was already beset by long layovers: four hours in Seattle, five hours or something like that in Anchorage. The flight from Anchorage to Sitka was supposed to include a quick stop in Juneau to drop off and pick up passengers, but that one took the longest time of all. Things seemed to be going smoothly on that last leg, though for some reason, the captain had to abort the first approach. A flight attendant let everyone know via the intercom that the captain would talk to us shortly, a little bit of an ominous-sounding announcement. After a few minutes he did come on the air and, in that casually reassuring tone that one gets accustomed to from the flight deck, apologized for the somewhat "botched approach." He casually mentioned that there had been a little problem with the landing flaps, and that we should be making another approach shortly. Once again we found ourselves above the clouds.
However, I noticed something weird. The landing flaps weren't out as far as they were supposed to be. Furthermore, while we were making a big circle around Juneau, the captain steadily kept up the nose of the plane at a unusually high angle. That's something you do in order to slow down the aircraft massively, as I had learned back when I used to spend time on my EASports flight simulator. A pilot with Air Alaska who happened to be seated in front of me confirmed that my suspicions were right. The flaps were stuck. He also mentioned that this was something highly unusual, which he had never encountered before in his decades of flying, except in the flight simulator. I was pleased to learn that he had made a safe landing on the simulator each time, and he seemed pretty sure that the present captain would most likely do the same, but somehow I didn't feel entirely reassured.
Nonetheless, we did get into Juneau safely. I think we came in just a little hot, but it was smooth, and--like I said--the runway was quite long. Obviously, we couldn't go on with that plane, and the lay-over in Juneau came to about 8 hours. There's not much to do in the gate area of Juneau airport, except to take some pictures of the gorgeous surroundings.
Finally, Air Alaska found another aircraft and flew all of us who were Sitka-bound to our destination. I made suret to listen to the landing flaps come out.
So, I got into Sitka in the evening rather than in the morning. Of course, at this time of year, there's not much daylight in between. The sun rose around 9 am and set around 4 pm.
I added up all of the actual time I had spent on the trip, beginning with leaving my house Monday afternoon to drive to the airport and walking into Wolf and Yvonne's house on Tuesday evening, and it came to 32 hours.
We'll pick up here with the next entry.
REFLECTIONS ON KING ASA
For quite a few months now, I’ve been contemplating the story of King Asa, as recounted in 2 Chronicles 14-16.
[I might mention that there was a time period when I was more or less the go-to person for B&H publishers on Chronicles, and I had the opportunity to write the Shepherd’s Notes and the Holman Old Testament Commentary on Chronicles, as well as the notes on 1 and 2 Chronicles in the HCSB Study Bible. (Perhaps I still am. We’ll see what happens with the new CSB translation. It’s been long enough that I wouldn’t mind getting back into Chronicles again—as I’m doing right now, I guess.)]
Asa was the 3rd king of Judah. In the past, there had been three kings over the entire land (Saul, David, and Solomon), and then the kingdom split with the northern “ten tribes” constituting the kingdom of Israel, and the southern “two tribes” forming a unit as the kingdom of Judah.On the northern side, the break-away king was Jeroboam, who was followed by Nadab and Baasha, while the southern kingdom had been headed up by Rehoboam, Abijah, and, now, Asa.
Let there be no question about the fact that Asa was a godly king. Quite early in his reign he found himself in a position where only faith in God would allow him to survive, and he trusted God. This event set the tone for most of his forty-one years on the throne of Judah.
Prior to the conquest of Palestine by the Hebrews, Egypt was in control of the Mediterranean East Coast. However, its army had drowned in the Red Sea under Pharaoh Amenophis II, and the country went broke under the money-wasting policies of Pharaohs Amenophis III and IV (Akhenaton--see my collection on him and his times). Thus, Egypt lost its supremacy over the Levant. From that point on, Egyptian armies would raid Judah and Israel from time to time, but could never again establish a solid foot hold.
The Invasion by Zerah
One of those attempted invasions came up when Asa was yet a young king; it was his fifth year on the throne. As various dynasties replaced each other in Egypt, the opportunity came for one Zerah of Cush (Ethiopia or southern Egypt) to enter the kingdom of Judah with a huge army. There is not enough information to know whether Zerah was a pharaoh or just a military field commander on behalf of the pharaoh. His army consisted of 1 million troops, including 300 chariots. There's a good chance that many of those soldiers were mercenaries, to whom combat was a way of life.
Short stack: What’s the single-most important issue I always bring up when I write about ancient armies? --- Time’s up; you’re right. Thanks for remembering. It’s the fact that they needed food and water: lots of it, every day. Pop!
Zerah and his million soldiers, along with the horses and chariots, would have drained available resources very quickly. I surmise that, to some extent, folks along the way must have cooperated with him. That is probably why, after the battle, Asa took measures against the facilities and the people in that area who had most likely assisted him.
It’s not that Asa did not have a rather jumbo-sized army as well. It consisted of 300,000 (3 lakh for my Indian readers) spear throwers from Judah, and 2.8 lakh (280,000 for my American readers) bowmen from the tribe of Benjamin. All of them, we are told, were also equipped with the right-sized shields for their roles. Thus, they were not just a rag tag band of conscripted farmers. A number of them may have had some experience, dating back to the temporary, but disastrous, invasion by Pharaoh Shishak about 15 years earlier. Also, some of them may have taken part in the ongoing minor border clashes between Reoboam, king of Judah, and Jeroaboam, king of Israel. (2 Chron. 12). Furthermore, about twelve years ago, during the time of Abijah, Jeroboam (with an army of 800,000 men) attempted to conquer Judah, and God gave Abijah (with only 400,000 defenders) a miraculous victory (2 Chron. 13). Therefore, at least some of Asa's soldiers probably had fought enemies before. Zerah’s army was almost twice the size of Asa’s, 580 thousand vs. 1 million, which included the dreaded chariots.
So. Asa and his army went to encounter Zerah’s mighty force “in the Valley of Zephathah at Mareshah,” a location that we cannot pinpoint with accuracy these days. It was somewhere south of Jerusalem in the Judean desert, close to the area still occupied by the Philistines at the time. He set up his troops for battle, but if either he or Zerah had any particular tactic in mind, it’s not known. The one thing that we do know is that young Asa felt overwhelmed by the situation and, believing that he was about to be defeated, turned to God and prayed:
“Lord, there is no one besides you to help the mighty and those without strength. Help us, Lord our God, for we depend on you, …” (2 Chron. 14:11a)
And God responded to that prayer in a miraculous way. There’s a rather telling phrase that the chronicler (Ezra for all that I know) uses several times on occasions such as this one.
"So the Lord routed the Cushites before Asa and before Judah, and the Cushites fled" (2 Chron. 14:12; cf. 13:16).
God fought the enemy in a rather one-sided battle, and Asa and his soldiers followed him. The reaction of fleeing when a battle is clearly lost is a common one, particularly if the troops consist to a large part of mercenaries. If you’re a soldier in a pretty sizable army, and you realize that your comrades right in front of you are being massacred, chances are very low that you’re just going to wait your turn and engage the overpowering enemy forces, just so you can lay down your life for king, brass, or country, let alone the promise of a pay envelope. It’s just human nature. You know you’re going to lose, so you vacate the premises. I understand that it’s very rare that one army or the other ever gets decimated on the actual field of battle. (One exception that comes to mind is Saul’s last stand in 1 Chronicles 10.) The terrible slaughters usually begin when the victorious army catches up with the fleeing enemy.
So it was here. Asa and his army pursued the hapless Zerah all the way south to the Philistine city of Gerar. I can’t help but think that Zerah’s soldiers weren’t exactly at the top or their physical strength by then. The Judahites plundered the city, got rid of the surrounding nomads who must have been supplying Zerah’s troops, and headed home.
Having returned to Jerusalem, a prophet by the name of Azariah son of Oded paid a visit to Asa. In the name of the Lord, he promised Asa that God was rewarding his faith in him, and that God would bless him and the country as long as they continued to trust him.
These events inspired Asa to undertake a great revival. There had been a lot of idolatry practiced by the inhabitants of Judah, who were supposed to be God’s people, and Asa set things right. He retired Maacah, his own grandmother (wife of Rehoboam), who had been a leader in promoting Canaanite idolatry. He removed the idols from the land. The king and his people made a pact that from now on they would worship Yahweh, the true God, and him alone. Also, Asa and his father, Abijah, had accumulated quite a bit of treasure, which he now donated to the temple.
Baasha's Evil Designs
All was going well from that point on until year 35 or so of Asa’s reign. Around that time, Baasha, king of Israel and a worshiper of idols, started to make hostile moves toward Judah. For one thing, he sought to block the steady trickle of his subjects into the southern kingdom. Furthermore, he set up fortifications near the border so as to have a base from which he could launch attacks into Judah. Then, to make sure that the escapades he was planning would come out successfully, he made a treaty with Ben-hadad, king of Aram, who lived in Damascus. Israel and Aram together would invade and conquer Judah.
Now, I’m going to continue this story from a point of view that makes sense to me, and you will notice that I’m bringing in some embellishments that I believe to be in consonance with the biblical text, but clearly not directly stated. If you are able to find a different perspective that leads to the same bottom line, that's fine.
Once again, things looked bleak for Asa. He needed to defend himself against an invasion of two armies who, in contrast to Zerah, would not have problems maintaining a supply line.
But this time Asa had a plan.
Now, if you ask me, it was a terrible plan, and you may agree. In fact, I think that at other times Judah's people would also have disapproved. But Asa was known as a godly king. He was a man of great faith who had led the people in the defeat of Zerah’s army. He had given much of his wealth to the temple. Under his pious rule the kingdom had flourished, and there had been three decades of peace. In short, Asa must have enjoyed the trust of his people, thanks to the outstanding faith he had displayed, and so we have no record of anyone challenging his decision until afterward. Now, this is my inference: I find it very plausible that, on the whole, people thought that Asa’s plan was not only ingenious, but yet another sign of his great devotion to God.
“Asa’s the man.” – “God continues to do great things through him.” – “We are so fortunate to have a king who is with us, the little people of Judah.” – "Who’s going to complain if he brings this off without any of us shedding a drop of blood?"
Asa made a treaty with Ben-hadad. He sent an official diplomatic note to the King of Aram, in which he alluded to a (quite possibly fictitious) treaty between their fathers, and “reminded” him of their friendship. Furthermore, in order to help Ben-hadad’s memory along, he accompanied the note with a humongous amount of treasure, clearing out his own palace as well as God’s temple.
The rationalizations must have gone on. "Isn’t that what the temple treasury is supposed to be good for, namely, to ensure that there is peace in the land for God’s people? Shouldn’t we see in Asa’s actions exactly how much of a servant of God he is? He had freely placed his treasure into the temple, and now he is relying on those donations to God to bring about a successful diplomatic solution for all people involved. Once again King Asa is demonstrating what a ruler in tune with God and people can do."
I, for one, have no problem imagining that people were thinking and speaking along those lines (except, of course in ancient Hebrew), from the high priest on down to the lowliest peasant in the field.
And Asa came through as only a man of his stature could. I mean, if there was any doubt about his machinations beforehand, the outcome surely must have demonstrated that he had been right in what he did. Ben-hadad gratefully accept the presents and agreed not to go to war against Judah.
I don't think that I''m too far off the mark if I imagine that some of the praise now went to the king of Aram as well. "Look at Ben-hadad. It didn't take much to sway him to the side of God and his people. To be sure, he worships idols on the outside, but inwardly, maybe he actually believes in Yahweh as well."
Now, here’s another thing about ancient warfare. Once you’re out in the field with a strong army, well-supplied and well-equipped, you can’t just tell them, “Sorry, the war is over. It’s time to go home again.” The troops are prepared to fight and expect to enrich themselves with loot after crashing the walls of a few cities. Ben-hadad knew this and immediately sent a message to his field commanders to attack cities of the northern kingdom instead of Judah. They complied. Needless to say, Baasha was very unhappy with this ugly double-cross, and he and his men removed themselves from the border area in order to fix things at home in the wake of the Aramean army’s attack. King Asa and his people proudly marched to the places where Baasha had placed fortifications, and tore them down, using the building materials for their own purposes.
Asa was the man of the moment. He was the king God’s people had been praying for. Another great victory had been won, and, once again, no blood had been spilled among the people of Judah.
Hanani and Asa's End
And then that prophet showed up. “Hanani the Seer” he was called. Was Asa expecting another commendation? We can’t say, but it seems like no one had yet called Asa to account for what he had done, and, so, he may conceivably have expected another divinely-inspired endorsement, just as thirty-some years ago from Azariah son of Oded.
But Asa got nothing of the kind.There was no praise for a cleverly designed plan that had protected Judah. Asa received no accolades for using the temple treasure for the good of the people. Hanani did not even offer any warning, let alone a bit of constructive criticism. What Azariah had said three decades ago still stood. Hanani simply conveyed a straightforward message of judgment. He reminded Asa of the fact that not he, but God, had won against Zerah and the army of Cush. If all the fiddle-faddle about Asa’s piety, as I have depicted it, had actually taken place, Hanani burst that bubble. For the rest of his reign, Asa would have to fight off enemies.
“Because you depended on the king of Aram and have not depended on the Lord your God, the army of the king of Aram has escaped from your hand. … You have been foolish in this matter. Therefore, you will have wars from now on.”
All the pretense that Asa had been basking in was gone. He had placed his trust in a pagan king, who couldn’t care less about God or Asa or the temple or human lives. His army’s services for destroying towns in neighboring countries could be bought for a price, and Asa had not only paid it, but used treasures right out of God's temple for the purchase.
The idea that no one had criticized Asa previously is borne out by his reaction to Hanani. Nobody talked to the king that way. Hanani was sent to prison.
Now a few other people started to grumble, and Asa responded by mistreating his subjects, ordering unspecified acts of cruelty.
Asa continued to slide backwards. He acquired a serious disease of his feet, but would not consult God by way of the temple priests or maybe another prophet. Instead, he put himself into the hands of the “physicians,” which was the word used for sorcerers who specialized in healing at the time (in other words, not medical doctors in our sense).
When Asa died, the people showed him all of the honors due to a great king, and—for a long time—he had been such. Only a few folks who had felt Asa's heavy hand towards the end were exempt from the peace and prosperity he had brought to his people.
Asa did well in his first crisis. In the second one, he entrusted the fate of God’s kingdom and God's people to his own plan and put the temple property at the disposal of a pagan ruler without a conscience. His subjects apparently did not care, though God certainly did.This story is very sad, and, as I said at the outset, I’ve been thinking about it quite a bit of late.
This entry is purely personal, just to get me back on track to regular blog posting.
Tuesday (2-28) was a little bit of a special day. When June got up she felt somewhat better than she had in a while, and she suggested that we go to IHOP for lunch, which was definitely fine with me. That would be the first time she'd been out of the house for a while, except for a couple of doctor's appointments. Furthermore, I happened to mention to her on Monday that one of those really tiny circuses was going to be here in Smalltown, USA, (how fitting!) on Tuesday, and she thought it would be a good idea to attend it that evening. She was right. It was a great idea, and we had a great time.
Now, before going any further with the circus and all of its intriguing details, I need to take a second to clarify June's current condition. When I'm talking about her not getting out of the house, that's quite literal. Yesterday, she used her super-duper 3-wheeled walker, complete with brakes on both side handles to navigate around, but on most days she doesn't really even have the energy for that little bit. She went from the house to the truck, from the truck to the restaurant, and then reversed the process. Same thing with the circus. From the house to the auction building of the local 4H fairgrounds and back. But, as minimal as it may have been, at least it was progress and a little sign of hope for improvement (not that her life is in danger, but her physical functionality is hovering somewhere not too far from zero.)
Towards the end of this month, she has appointments coming up with the neurologist and the cardiologist. As it stands, her diagnoses are a serious case of fibromyalgia, accompanied by chronic fatigue and sleep apnea. The appointment with the cardiologist will be the first follow-up to the tests she took way back in December, which may strike you as mismanaged as it does me, and that's all I'm going to say about that matter at the moment. At home, June is usually in her recliner, and I find myself in the role of food fixer, at which I do okay, and cleaner-upper, which is not my strength. It's a point of pride for her to continue to do the laundry, about a load a day, and I'm glad that we go the new washer dryer set a while ago because it's a whole lot easier on anyone's back.
I've been hanging in, of course, but my regular depression has been amplified by the current circumstances. So, I've been in something of a malaise, trying to undertake a few serious things, but finding myself doing stuff that's perhaps not totally insignificant, but doesn't carry a lot of pressure. Specifically, I've been spitting out a lot of answers on Quora. Many of them have had to do with English grammar, which is usually neither horribly controversial nor difficult to deal with. Writing comes easily to me, and intellectually a game of Free Cell or Sudoko is more demanding than explaining subject-verb agreement. I have usually announced my responses on Twitter, but not all of them.
I have not been entirely unproductive in matters of a little greater import. For example, I have reworked the beginnings of my on-line course, Sanskrit 101, and fixed it up so that it is now available to anyone who might be interested in learning that valuable language. So far, it only goes through chapter 4 out of 14 for this first course, but I'm sure I'll be ahead of any students in getting the material up. Please be aware that my next step will be to reduce the introductory verbosity.
Speaking of such things, I've been hoping to do another Sanskrit session at the ISCA conference in Texas at the end of this month, but I haven't publicized it up to now because I'm still not sure that I'll be there. It'll depend on what will develop with June's situation. At the moment, it would be risky to leave her home alone.
Over the last few months I have been able to maintain my weekly StreetJelly show most of the time. There were a couple of Thursday evenings when I just couldn't get myself together, and for a few weeks I was fighting internet issues, but the show has gone on for the most part. In fact, for the last couple of weeks, it's been really fun again.
So, I'm planning on doing my usual show tonight (Thursday). The theme is "All Originals," and it'll probably come to you from the Prancing Pony in the town of Bree at the edge of Mirkwood. June has expressed the need to be somewhere where she can spend a lot of time in the roiling and boiling waters of a jacuzzi, and I readily assented, seeing that I always like to be out there in deep wilderness beyond the walls of Smalltown, USA, our counterpart to the Shire.
So, to return to the circus, I was particularly intrigued by the little flyer I picked up in the drug store on Monday, announcing that the Stardust Circus has one of the few remaining circus elephants. Let me clarify. Prior to announcing that they were shutting down, the Feld Family, heirs to Messrs. Barnum and Bailey, not to mention the seven Ringling Brothers (Alf, Al, Charles, Otto, Gus, Henry, and John),* had been under pressure to retire their elephants. So, I was curious to see what the Stardust Circus would offer in the proboscene department. Well, they had one extra-large African savannah elephant. Prior to the show and during intermission, he provided rides for kids around the small arena. His performance in the program was quite low key, basically swaying (i.e. "dancing") to music. The behaviors such as raising one leg is something that every elephant under human supervision is taught because it makes foot hygiene possible. (And, just in case you're wondering: No, I did not spend 10 dollars to sit on an elephants back for three minutes. I find elephants to be cumbersome and awkward to ride unless you get the good seat right behind his head. I still hope to make it back to Thailand and get a real elephant ride there.)
Admission to the circus was free for children. Of course, the entrepreneurs collected the equivalent amounts easily in elephant rides, plastic toys, light sticks, mouse ears, nachos, drinks, and snow cones, and how in the world can I forget those pink and blue clouds of cotton candy? It's amazing how much money parents wind up spending to keep their children from feeling left out. Not that I blame them, and by "them" I mean both the youngsters and their significant adults.
June and I were sitting on the front plank--"bench" if you want to stretch the term to include planks nailed onto the occasional post. We were probably the only adults there without little ones, but, since I'm still such a child when it comes to circuses and such, I did not feel particularly conspicuous, except for a few brief moments that I'll tell you about shortly. I might mention that I didn't have my camera along, so the few pictures that even came close to useful came from my flip phone.
Come show time, the sales staff turned into the performers. It seemed as though they were all related, hailing from some corner of South-East Europe, which I could not identify any further. The middle-aged, somewhat heavy set, lady who sold the nachos turned into the beautiful Someone (I didn't catch the name), who balanced flowers, dish sets, and swords on her chin while she climbed up and down a ladder. The gentleman who had sold the light sticks performed tricks with spinning tops of various sizes and served as the one and only clown; he was responsible for the moment of conspicuity which I'm saving for a little later in this account.
The most talented of all the show folks was the young man who had sold me a snow cone earlier. He was the center of three performances: 1) juggling clubs, rings, and hats, 2) holding his balance while standing on a stack of boards, each one of which was separated from the one be below it by a rolling can, and 3) doing clever unicycle stunts alongside his brother.
Other than the elephant, the only animal present was a lovely pony who walked a slalom line between traffic cones and raised his forelegs together in salute. I guess one can think of him as a "two-trick pony." Then again, given how close-up everything was in the barn, everybody could see how cute he was, and I think that was the best part about his act for everyone.
As to the clown, he made two appearances. Due to the need to change quickly between activities, he never wore full clown attire. He had on a different shirt, a red hat of uncertain shape, and a very slight amount of clown-like make-up. In his first appearance, he came out with a whistle in his mouth, on which he blew incessantly and got the two halves of the audience, on opposite sides of the small arena, to clap in rhythm with him and each other. Then a girl in her early teens came out with one of those large cardboard fan-like slap sticks, designed for maximum noise and minimum effect, spanked him on his south end, and told him to keep quiet. The plot developed along predictable lines. Neither June nor I thought that the humor rose to its full potential, but the children around us sure loved it.
The second clown act also took place before intermission. He began by tossing potatoes up in the air and catching them with a fairly large two-pronged serving fork, the kind one uses to get a joint of roast beef out of the oven. Having demonstrated his skill, he now handed a potato to someone in the audience who lobbed it at him, and he speared it. He did this for about a dozen times, ranging over the various areas of the bleachers, usually with success. I think I was the third or fourth person, and I must confess that I may have been responsible for his muffing the play. Despite my best effort, I probably did not put enough of an arc into my toss.
A few minutes later, when it appeared that his act was about to come to a close, he trudged back to me. This is the point when I began to feel somewhat conspicuous. Of course, if you know me personally, you know that I don't have a problem with being conspicuous, and that, when it comes to communicating in "clown," I'm pretty fluent in that language. He handed me another potato and backed up a littler farther than the previous time. I felt as though my arm and aim came through for me that time. But no. The vegetable slid right off his fork.
No, wait! I got another chance.
He handed me yet another "potato." I would like to go on record that my throw was perfect that time.
Whoops! What happened now?
The sly clown never even stuck out his fork, but let the potato bonk him on the hat. Everyone laughed, and I watched with amazement as he strode back to me one more time, reached in his pocket, and handed me a little sour-apple flavored sucker. I held it up and doffed my hat as he made his final bow to the applauding crowd.
The message of the sucker was, of course, clear. As David Hannum said in observation of the folks who were shelling out their money to see P. T. Barnum's exhibit of a fake petrified giant, "there's one born every minute." (It is now generally accepted that Barnum was not the originator of that celebrated phrase.)
If I sound as though I'm promoting Circus Stardust, I guess it's pretty close to that, though obviously not for any reciprocal considerations. Which leads me to one final note: The whole show was geared to young children, deserving a G rating. It contained no suggestive matters, nor any foul language. How rare is that nowadays, even for so-called "family entertainment"?
That's it for today. Just some purely personal things that I'm trying to use to get back into the pace of doing my blog regularly.
Next time some reflections on the topic: "Kings and their Wars: Where do You Look for Help?"
*There was also a Ringling Sister, named Ida.
June’s Adventures in Hospital Land
Enough people have asked how June's tests went, that I'm writing a single narrative here. Thanks for your concern. Nothing really happened, but, as many people know, I enjoy giving my P.G. Wodehouse-inspired side a chance to show off. The really important thing will be the results, and we have no idea when we'll hear anything. Much of what I’m writing here is based on June’s ongoing reports to me, and I’m pretty sure I remember everything accurately. However, she is not responsible for any hyperbolic descriptions subsequently forced on me by my irresistible need for creative expression..
First, she somehow disappeared through a mirror, and then she met this really big white rabbit. And when she saw Humpty Dumpty sitting on a wall, things really got weird. No, wait … Sorry. Different person, different events.
Obviously, June’s condition has not changed because nothing has yet been diagnosed or treated. Yesterday (Tuesday) and today (Wednesday) we were at the hospital for cardio testing: a cardio-echogram and a Cardiolite radioactive stress test. In case you’re interested in some of the chemical aspects of the latter, I put together a summary about it in my blog from May of last year, when I had that test (and passed it with flying colors). It’s a two-day test, consisting of an injection of radioactive 43Technecium (99mTc), a CT-like nuclear tracing on day one, followed by more injections, a treadmill test, and another session with the nuclear folks on the second day. The cardio-echogram was supposed to happen after the radiation test on the first day. Sounds relatively simple. But …
Day 1: After a short time in the waiting area, June was called into the cardio quarters and injected with the radioactive brew. No, she didn’t glow, but as I said on the blog back then, if there had been a Geiger Counter around her, it would have been ticking. Right after the injection, the ultra-sound lady, apparently not acquainted with other items on the schedule, swooped down on her and proceeded with the cardio-echo. June has had ultra-sounds before, but it never had been painful before. This lady apparently was just a little abrupt in her manners, and her technique was not exactly gentle, as she virtually stabbed her wand between her ribs. (I warned you about hyperbolies.) Afterwards, she told June she could go home.
So, June came back to the waiting area where I was huddling and started to put on her coat. Fortunately, I knew that the radio-active test was far from finished, so, after a couple of minutes of all-around confusion and a teensy-weensy bit of assertiveness on my part, things got resolved. Not too long thereafter, we were off to the nuclear medicine dungeon. This time I went in with her, not for any reason other than to keep June company and because I like that creepy place. It’s actually a perfectly normal x-ray-looking facility, except that there are bodies lying motionless on cots surrounded by technological apparati. The scene reminded me a little bit of "The Matrix" or other films with similar components.
June went through her immobile period in grand fashion. Parmenides and Zeno, the ancient Greek philosophers who asserted that there was no such thing as motion, would have been proud of her. When she was done, the nuclear medicine technician came over and confessed that, alas, a piece of her intestine had blocked the machine’s view of her heart.
Lunch time. We returned to the nuclear reactor, woops, I mean department. This time everything worked. Home. For someone who is suffering from extreme exhaustion, unsurprisingly, June was extremely exhausted.
Day 2: Today started with some more injections and the treadmill test. The friendly woman who served as treadmill administrator informed June that she would just spend a few minutes walking on the rolling band, which would increase its incline every minute. Just a short time earlier, when we had entered the hospital, June had some difficulty walking up a slanted hallway, needing to stop several times to catch her breath. (Don’t ask me why there is such a thing as an uphill hallway in a hospital.) So, June responded to the treadmill lady that she was certainly happy to walk on the treadmill, but that she couldn’t guarantee that her knees wouldn’t buckle in the process.
There is a standard alternative. Instead of walking on the treadmill, the patient’s heart rate can be elevated chemically by way of an adenosine injection. After some consultation, it was decided that they would do this with June. Apparently, it was not totally routine, though, since the room was suddenly filled with personnel from various departments, just in case something would not go as hoped for. June did not show evidence of a bad reaction to the adenosine, and everything went fine, not as they had planned, but as I had figured.
Now back to the nuclear department for another round of motionless scanning. Well, make that two rounds, with a time for lunch in between. Once again, June’s insides blocked the camera’s view the first time, but settled sufficiently downward the second time around.
That’s it. Nothing really dramatic, just some unexpected little contingencies adding interest to what could have been a purely routine set of medical tests. Now we're waiting for the results.
I'm afraid I don't think I can make this little post without it looking like an ad, thought it's not intended to be one, at least not in the commercial sense. In the process of my never-ending series on phi, Dan Foley and I crossed paths and discovered that we both shared a love for learning. He is an artist who recreates ancient Celtic decorations, and on his web page he explores some of the meanings of the symbolism behind it. It's an intriguing site. I'm going to put a long-range link on here somewhere, as I'm about ready to start a new series (topic suggestions still welcome). For now, this little entry will have for its sole purpose to draw attention to Dan and his work. Please visit the site of D & O Jewelry.
I really had no intention of writing yet another piece of math for the blog. As I was going through my e-mail around noon, I came across some notification from Quora.com, which eventually led me to the question:
If you should look it up, you may agree with me that the answers currently in place pretty much assume that the person who asked it is already familiar with calculus and that comments get pretty esoteric pretty fast. So, I thought I would write a quick answer that would be intelligible to someone who didn’t already know calculus. Well, seven pages later, I hope that I have possibly helped someone understand the basic nature of calculus a little better without totally scaring them off. However, my answer is quite a bit longer than anticipated (as always), so I will post it on my blog instead in the hope that someone may see it and read it. I am trading possible “up-votes” for potential intelligibility.
There are two parts to calculus. One of them is “integration,” and the other one is called “differentiation.” I will mention differentiation in passing in order to highlight what integration is, but I’ll make sure that I won’t stray too far on a tangent. (Wow! Sorry, that pun was not intended.) The question is one about the meaning of a term in mathematics, so my explanation is not going to include proofs, lengthy calculations, or application of complex formulas, though I may reproduce a few just as visual aids. I’m going to use two illustrations spread over three sections. Section 1 consists of a basic explanation of integration; section 2 brings up a badly needed reality check; and the third section is something totally different. I would suggest that if you don’t want to follow the whole exposition here, but you want to get something out of it, you might want to go straight to section 3.
(1) I can’t imagine why I would want to buy a flock of sheep, but let’s pretend that I do, and I visit a man who has kept sheep for a several decades, but wants to retire now. As I look over the entire collection of wool bearers, I realize that there are too many of them for me to count, not even if I resort to one of the standard tricks, such as counting all the legs and dividing by four. So, I ask the shepherd how many there are. He replies, “There are about a thousand sheep now. I started out twenty years ago with just ten, but the flock has grown enormously.” (The story is getting less and less probable; a shepherd would surely know exactly how many sheep he has.)
Having received that number, I want to mine it for some more information, and so I set up an equation that relates the number of sheep to the 20 years. The number of 10 for the original sheep does not change, and I will simply add it as needed after basic calculations. I’m thinking of dividing 1,000 sheep by 20 years, and come up with a rate of 50 sheep per year). I write out an equation:
number of sheep = number of years × 50 + 10
s = 50y + 10,
where s stands for the number of sheep and y for the years.
So, after 5 years, there would have been 260 (5×50+10) sheep and 1010 (20×50+10) after 20 years. Five years from now I’m expecting to own around 1260 (50 × 25 +10) sheep, a manageable number. [If you are catching a bit of a misunderstanding on my part, please follow me here and wait for the next section.]
I’d like to visualize those numbers, and I make myself a graph that plots the number of sheep against the years. Since I don’t intend to reverse the flow of time, and since I don’t have any idea what a “negative sheep” would be, I can stick to positive numbers.
This a nice straight line containing no surprises. I extended the line to 25 years since I want to see what to expect in the future, but I dropped a vertical line down to the 20-year point because that's our present location in time.
Even though we already know the rate of change, allow me to add a little bit of terminology. We can understand the equation as a function.
The rate of change is called the “derivative” of a function. There are various ways of writing this out. Because differentiation is not the point here, I’ll just choose the simplest one. The derivative of function f is f', and thereby
Since the line is straight and we already knew that information anyway, we’ll just leave it at that and move into the other direction, which is integration.
I know how many sheep I have and at what rate the flock grows, but I’m a little afraid of how much work will be involved. For example, I’m wondering about how often I will have to shear those sheep, not just in one year but over many years. I know that I should do so halfway through every year. In order to make this this calculation easy for now, let me make the somewhat bizarre assumption that the growth of the flock occurs continuously over any given year. There will be a new sheep in my enclosure every 7.3 days, which translates into 7 days, 7 hours, and 12 minutes. We know it doesn’t work that way, but let’s just keep it simple. The point is not the number; the idea is to avoid unneeded complexity and think of the equation as an unbroken continuous line. We’ll have to get back to this assumption soon.
Given these two considerations: (1) the continuity of the function and (2) the need to shear the sheep halfway throughout the year, let’s figure out how many sheep our friendly shepherd and his helpers must have sheared so far. Since we have just finished the twentieth year, the last shearing was ½ year ago, at 19.5 on the year axis. Now, it might be tempting simply to plug in 19.5 as y and calculate
50 × 19.5 + 10, which comes out to 985.
But then we’ve merely calculated the number of sheep that were sheared on the last occasion. I want to know the total amount over all of the years.
Now we are moving into the area of integration. To put it briefly we want to calculate the area underneath the line of the function.
Given the simplicity of our example, it’s not too hard to calculate the number the numbers for each time (e.g., at 0.5, 1.5, etc.) and add them all up. There is no great need for calculus and integration here, but that’s because I’m using a simple example to illustrate the method. There are many far more complex problems for which a simple count-and-add procedure is not possible. We can get the information we’re after by dropping a vertical line at 19.5 and then calculating the area of that triangle. There are several methods with which we could do so.
One would be to try to figure out the area of the triangle, given whatever values we have and plugging them into the formula. A more generally useful method in the long run is to apply a formula for “indefinite integration,” which runs in its most basic form (for exhibition purposes only for now):
The integral of the equation above = the formula on the right
where C stand for an undetermined constant. I'll plug in one set of values so you can at least make a bit of sense out of it. Let's say you have the formula
Then its integral will be
As promised, I’m not going to apply any really complex formulas or do any long calculations. Besides, I have a huge problem on my hands all of a sudden. The shepherd is laughing his head off.
(2) The shepherd has glanced at my graph and, after just a short look, seems to think that it’s the funniest thing he’s ever seen. “It doesn’t work that way, Win. We’re talking about living animals. A flock doesn’t grow along a straight-line graph.” I get it. Some sheep get sick and die; some may be sold; others may be stolen. One can add to a flock by buying more sheep as well as having them make lambs. If they all would reproduce according to the proper mathematical pattern, their growth would geometric. In the real world, that would not the case for many practical reasons. However, if we looked at their growth pattern in an ideal setting, the sheep would multiply by an exponential curve. In plotting this graph, I had to settle for a relatively slow rate of growth in order to squeeze the growth from 10 to 1,000 into 20 years, but it’s still going to make the difference between the two approaches very clear.
Here is what a graph for what this equation looks like:
The geometric form is that of a parabola; not only does the number of sheep grow every year, but the rate at which it does so gets faster. The longer it takes, the steeper the curve gets. What starts out as almost a horizontal line eventually will be almost vertical. Because of the example I chose, I had to make this graph come out at 1,000 just like the previous one, and If you remember the straight-line graph, you can see that up until that point the parabolic line actually runs underneath the linear one. However, given the nature of parabolic lines, there must be a point where it exceeds its linear counterpart, and it’s right about here. It cannot remain below the straight line forever. In another 5 years, I will not have merely 1,260 sheep, but 1,598. And looking at ten years from now, the straight-line curve would only give me 1,510 sheep, but under the exponential growth curve, I would have 2,250. The rate of growth is slow, but getting faster and faster. I’m not entirely sure about the numbers that Wolfram-Alpha has given me after my somewhat quirky request that the first 20 points on the x-axis had to stay below 1,000, but they make the point without my having to fit in a graph that wants to zoom toward the sky so fast that the lower section becomes unreadable.
If we want to find the integral in order to figure out how often the shepherds have sheared the sheep so far, we have to look underneath the line again. This time we’re not getting a nice right triangle. We need to use the integration formula, and I’m keeping my promise not to make you slog through line after line of equations, but obviously these numbers will become much higher than those of the straight-line function as well.
(3) Okay, everybody (almost?) knows the two celebrated formulas concerning a circle. The area of a circle is calculated by the equation
and we get the circumference of a circle with
Here’s what I would like you to see because here we have both integration and differentiation in action. π is a constant that we can ignore here for our demonstration purposes. If we want to get the integral for a circle, that’s kind of a tricky business. There is no genuine function under which we can calculate the area. However, we can calculate the area of the inside of the circle. That procedure is analogous to finding the integral of true functions, and it's easy to remember as long as you know the two circle formulas.
In this case, take the formula for the circumference, 2πr, make 2 the exponent of r, and divide the formula by 2. You get: πr2, which we immediately recognize as the formula for the circle’s area. It is also the analog for an integral.
Conversely, the pseudo-derivative of a circle measures the length of the curve surrounding the area. We get it by taking the exponent 2 from πr2 and making it a coefficient, 2πr. Voila! There’s the formula for the circumference again.
Since, as I said, the graph for a circle is not truly a function in the technical sense, these are not actually instances of differentiation and integration. However, they are close, probably easier to remember, and a good introduction to these two sides of calculus.
I think I have successfully answered the question on What is integration in math? My answer is way too long to fit into that forum, but I really wanted it to be of help to someone. Not wanting it to go to waste, I made it a blog entry. Hopefully, people who are interested in such a question will see it and find it helpful.
If you read this entry and followed my line of thought, and you’ve never studied calculus before: Congratulations! You are now an initiate on the very first rung of that mysterious part of mathematics.
It took a while longer than I thought, but I finally have the website on Phi and the golden ratio in order. It's now totally reordered from the original blog posts, with the different sections combined and divided in a way that make sense. There are more links to serve as bridges to let you jump over fierce-looking equations. Hope someone will find this material helpful.
This entry has nothing to do with math.
Nor theology, nor music.
Nor world religions.
In fact, it doesn't have to do with much of anything except a funny comment made by Tim H. on Facebook in response to one of my postings on my secret identities or deep personality structures. I wanted to respond to him with a funny cartoon that I thought Gary Larson had drawn in his famous "Far Side" series. I never did find it. It was a picture of a van, which was nicely labelled "Witness Protection Program" on its side and, if I recall correctly, the vehicle was being attacked by some mobsters. The driver was shocked that they had been identified--or something like it. Maybe it was not Larson's. It could also have been been in the Atlantic.
Anyway, in the process of looking for that one cartoon, I wound scrolling through a few hundred cartoons and pictures, and, if you should see me over the next 24 hours or so, and I suddenly break out laughing, it'll probably be because I just remembered one of them. Well, it occurred to me that, even though I didn't find the one I had been looking for, some of the others were just as good and definitely worth sharing as a blog piece. Kind of a break from my recent fare. So here goes. Enjoy. Rejoice with me that maybe, since the last couple of years more or less, my blog won't give you a headache. The pictures are all "click-throughs" to the websites on which I found them, but any owners or originators who wish that I remove their creations will find me pleasantly agreeable.
Witness Protection Cartoons
Until next time!
We just went through a tornado warning about as close to us as I ever want to get. Thankfully the bad part of the storm decided to move about a mile north of us.
If you would like to get caught up on all of the previous installments of this series, please do so by going to the website on which I have been collecting it: PHI—Let’s Get It Right. I’m putting in corrections, and pretty soon navigation aids, there, rather than backtracking on the original blog entries. Some of the sections will be totally rearranged so that the sequence makes sense and some duplications can be eliminated.
This has been a long mega-series, and no one is as surprised by its length as your fatigued bloggist. The initial motivation was a pretty modest one. As I stated right at the outset of this series, my point has been to introduce some cautions into how much use a Christian apologist may make legitimately of the golden ratio, the Fibonacci numbers, and phi (ϕ) in an argument for the reality of God. It has not been my goal to give a full description of the history of phi or of all of its occurrences, let alone the Fibonacci series. Still, I wound up adding a whole lot more material than I had originally intended since I realized in the process that the need for analysis was greater than I had anticipated, and that any number of items would not make sense without providing at least some background.
Having occupied myself with these things for several months now, I’m thinking that the phenomena connected to phi or the golden ratio are best combined with other features of the world for the sake of an argument for God’s existence. Still, the fact that phi shows up in so many different and apparently unrelated places surely is remarkable. As you have read, I am even more amazed by its intrinsic properties as they emerge in geometry and algebraic equations than in their occurrence in nature. Consequently, as a believer in God, I do see his handiwork as an all-knowing and all-powerful creator in this aspect of the world he made. When I say “world,” I’m referring to both the physical world of matter and the world of numbers.
However, it is in the much broader context of mathematical and physical reality that the evidential side of the subject matter becomes strong. Phi is a number; in fact, it is a unique and truly remarkable number. So is pi, though in different respects than phi, and so is e, a number whose characteristics would take too long to explain here. Let us not forget 0 (the “additive identity constant”) with her many characteristics, or 1 (the “multiplicative constant”) to whom we did not pay much attention in this discussion. There is i, the square root of -1. There are so many kinds of numbers, as represented by their sets from ℕ through ℤ, ℚ, ℝ, to ℂ, as well as special sub-categories, such as ℙ. When we come to terms with the complexity of the universe, while simultaneously recognizing how mathematically fine-tuned it is, it becomes impossible for me to say, “It just happened."
The evidence for God is there. Unfortunately, some people confuse the creator with his creation and think of the created order as God. However, nothing can be the cause of its own existence and, if there are indications of intentional regularity in the universe, they reflect on the one who made it. When I think of various non-theistic theories of the origin of the universe, I can’t help but wonder how anyone can be satisfied with them. For example, take Michio Kaku’s claim that our present universe is the result of a collision of two previous universes. This idea not only strikes me personally as bizarre (which by itself does not count as evidence against it), but leaves all of us hanging with the question of what caused the existence of those previous universes. Another set of colliding universes? And another one before them? Are we back to turtles supported by turtles?
Most important for this series, though, has been my intention to encourage Christians to test the evidence before using it. I have been quite critical of the unbridled way in which many people have attempted to “find” the golden ratio in various parts of nature, architecture, and art. For many of them, the need to find phi in virtually every beautiful building or painting, seems to be driven by a mystical outlook that directly links beauty and the “golden number.” As far as I can tell, a direct causal relationship between phi and the perception of beauty has neither been completely undermined nor proven. However, there is definitely no necessary relationship between beautiful objects and numerical patterns embodied in them. In many cases where the Fibonacci numbers appear, e.g., among flower petals, their purpose appears to be practical rather than aesthetic. This assessment does not take away from its wonder or from its contribution to the evidence for a Creator, but it does extend our understanding of phi beyond creating visual appeal.
Phi is a number. More specifically, it is a number that arose out of the relationship between various geometric lines found in the pentagon. It belongs to geometry, analysis, and number theory. It is not something that we merely run across in our day-to-day existence.
Consider the example of the bones of a finger, which we mentioned earlier under the heading of “Phi in Nature, part 3.” Why are we interested in their relative lengths? If, indeed, they do measure 2, 3, 5, and 8 units in length, that discovery per se doesn’t tell us much. 8 divided by 5 equals exactly 8/5 a nicely rational fraction of two integers, which can also be expressed as 1 3/5 or 1.6 with no further remainder. There doesn’t seem to be anything special about it in isolation. If we were ignorant of any further implications, such as the Fibonacci numbers or phi, we might write down the results of our measurements, but we would probably not excitedly post them on a website to share with the world.
The reason why we are intrigued by the lengths of the finger bones is because we already know that these numbers have a larger significance: they are a part of the Fibonacci series, which converges to phi. We fuss over these numbers because we anticipate what’s coming. When we do nothing but measure the proportion between the proximate phalanx and the metacarpal, there is a huge difference between the result we get,1.6, and the number we would eventually reach if we could go on forever with the measurements, namely ϕ. We do not see ϕ exemplified on the x-ray, and our limited data would not allow us to compute ϕ. However, we have learned previously that the Fibonacci series will converge to phi (1.618033…). Thus, we find the Fibonacci numbers in nature and give them significance based on the Fibonacci series, but the series is actually a mathematical entity. It is the outworking of a recursive equation,
Fn = Fn-1 + Fn-2
with start-up values F1=1 and F2=1,
not a principle emerging from the observation of human digits or fecund lagomorphs.
So, what am I saying? Nothing that I haven’t said many, many times, but ever more frequently over the last few years. Apologetics is not about memorizing answers and arguments. We can leave that approach to the internet atheists and their irrepressible need for rabbit trailing. It’s about finding answers to relevant questions as a part of the total project of demonstrating the truth of Christianity, both in evangelism and in dealing with our own doubts. And in that context, memorized answers (“If they say this, then you say that”) without a personal understanding of both the question and the answer, are of very little help, if any.
I’m afraid that phi is a case in point. It’s easy to list it as one of the marvelous aspects of the world that points to the Creator. But I need to ask, to what extent can you substantiate the make-up and meaning of the golden ratio? Can you discriminate between what is science and what is pseudoscience? What is nature and what is numerology? What is truth and what is fantasy or deception? I’m hoping that this series has helped create a little more understanding and, perhaps, even a little bit of interest to delve further into this topic or the role of math in apologetics.
As you have seen, I have built bridges by means of hyperlinks into some of the blog entries and even more into the combined site so that you can jump over some of the meanest-looking equations and calculations. It’s a great feeling when you learn to work through an equation and actually get the result you’re supposed to, but doing the math in all of its fine points if not obligatory. Nonetheless, I do want to say that anyone making use of the remarkable number phi should have some basic understanding of:
1. the difference between phi and the Fibonacci series;
2. the fact that phi is not derived from the Fibonacci series, but that the Fibonacci series converges to phi;
3. the nature of the proportion (whole to large segment = large segment to small segment) that constitutes the golden ratio;
4. the fact that phi originated in geometry;
5. the basic nature of a golden triangle, rectangle, and spiral;
6. some of the genuine occurrences of the Fibonacci numbers in nature, art, and architecture;
7. the lamentable fact that people do fudge the data in their apparent eagerness to find the golden ratio everywhere;
8. our obligation as Christians to be honest and forthright in our learning and speaking. It’s not right to use bad information, even if the person to who you are talking believes it. (Nor, for that matter, is it okay to make up something on the spur of the moment if you don’t know the answer, not even if you think you just experienced a moment of revelatory inspiration [or inspired revelation].)
9. my concern that in order to become a good Christian apologist, one should study fields of knowledge as a whole rather than just picking supposedly good apologetic tidbits out of them.
10. the happiness that results from studying and learning!
It’s August. The temperatures are cooling off, and the days are getting to be rather nice instead of sticky. June is hanging in health-wise, but not doing as well as she should. Let me just relate the minimal story, if you don’t mind. She is showing some serious nutritional deficiencies, which means in her case that the body does not easily absorb them even when their present. The condition may be resolved with megadoses, but the question for us is also what the necessary supplements will do to her digestion. (As for me, I’m pretty good for the most part.)
We had a rather active weekend, by our standards. On Saturday we spent most of the day at a nearby resort/camp ground, though not to return to nature (as much as we want to do that again, given the next opportunity). We were a part of a gospel sing on behalf of MDA, the Muscular Dystrophy Association, which was held in a big “commons room.” It was good to get together again with some of my friends from “Cowboy Church.” There were a number of bands and some soloists. The quality was variable, of course, though nobody was really bad. I did one set in the afternoon, right after which I accompanied Susan and Bob, former fellow members of the Tippyditch Singers and Cowboy Church performers, on my bass.
It was my first time playing the bass publicly in well over a year, as well as finally using the new amplifier I received Christmas before last. So, that song was my warm-up, as it were. Then, later on in the evening, I played bass along with another friend. By then I felt quite comfortable on the instrument again and I confess that I surprised him a little bit when, out of sheer high spirits, I inserted some lead lines.
Then yesterday was Jellypalloza, the celebration of the fourth anniversary of StreetJelly.com, one of those days on which all regular StreetJelly artists are invited to do a short set. I did mine in the early afternoon, wearing the recommended tie-die look. My program consisted of a more or less random selection of songs I like in the folk, country, and gospel categories. This coming Thursday night is the fourth Thursday of the month, which I usually designate as an all-gospel night. My customary time is 9 pm Eastern, and—as always—it will be live on the internet at StreetJelly.com.
Last night we got together with the “boys” and their wives at Sitara Indian restaurant in Muncie to celebrate my birthday. I have totally fallen in love with their lamb biryani, which they prepare Hyderabad style.
Where else in nature can you find the golden ratio?
One suggestion, advanced by “Mr. Phi,” Gary B. Meisner, is in the proportion of bones in our arms and hands. I have referred to some of Meisner's expositions in other contexts several times already. He is definitely a phi advocate, but a critical and honest one.
Let us look at a couple of instances. I have approached them with as much skepticism as I could reasonably muster. Still, by the time we're done, it looks as though the proposed proportions are present, at least in acceptable approximations.
First, here is Meisner’s depiction of an arm with a hand:
The golden ratio in the human arm and hand,
picture by Gary Meisner from his site.
I went through my usual process of measuring the pixels of the various lines, and here is what I found. Remember, that the dimension of the picture depend on your screen size, magnification, etc. But the proportions should hold true within technically possible parameters.
|Hand||136 pixels||Forearm to hand: 1.59 to 1|
|Forearm + hand||352 pixels||Forearm + hand to forearm: 1.63 to 1|
Those numbers, i.e. 1.59 to 1 and 1.63 to 1, are certainly quite close within the inevitable margin of error, allowing for fallibility in my measurements as well.
I’m not sure about the placement of the white line separating the hand from the arm. It seems to me that it may be a little too far to the right, past the wrist and cutting off a little bit of what I would consider to be hand rather than arm, and if we moved it a little further left, the ratio would go down. Measuring my own arm and hand did not come up with similar results, but pretty much stuck to 1.5 to 1 even with the dividing point where Meisner put it. I'm not saying that Meisner fudged, but there does seem to be a disparity in my judgment vs. his.
Then there is the proportion of the various little bones of the hand and fingers to each other. We are looking at four components for each finger: three phalanges and one metacarpal, all of them rooted are in the set of carpals.
Diagram of the bones of the hand
by Mariana Ruiz Villarreal (LadyofHats)
Wikipedia Public Domain
Here is Meisner's x-ray view of one forefinger.
The bones of a human forefinger.
Picture by Gary Meisner from his site
As you can see in the picture, the size of the little bones follow the Fibonacci numbers more or less. Let's agree on more than less. I’ve copied the picture and sunk the lines from the ruler down across the pieces, so as to in order to enhance our ability to visualize the approximations.
It appears that, given a bit of leniency, the Fibonaccis are there: 2, 3, 5, 8.
Still, I do have a little bit of a problem with this display. I don’t have an x-ray of my fingers handy, and I can only measure my own finger externally. But that fact doesn’t mean that I have no way to test what Meisner is saying. The fingernail is supposed to represent one unit of length. I can’t measure the bone underneath my flesh and skin, but I can measure how many fingernails long the distal phalanx is, and that comes out pretty closely to two units, by assumption and some observation. However, measuring the second section (the intermediate phalanx) does not get me close to three by that scale. At first glance distal and intermediate bones seem to be equal; depending on how I position the finger and ruler, I can get it up to 2.5 fingernails long, but that’s all. Thus, if you use my finger as in any way representative and allow for external measurements, the Fibonacci series is not quite as prominent as one would like. Then again, my fingers may be oddly proportioned in various ways. (Speaking of which, I’m happily looking forward to my trigger finger surgery, still a couple of weeks or so away.)
The lesson is simple, but important. The regularities of nature, even when they manifest a mathematical formula, do not necessarily conform to it with complete mathematical precision. God did not create assembly line robots, but individuals. Theoretically, one could say that each individual's departure from the mathematical template is a flaw or defect. But it's the departures from the theoretical blue plan that makes us individuals, unique, and worth knowing.
It's still early August, but the local schools are already in session. The weather is hot and humid; I wonder if those kids are learning anything. Or, let's say, if they're learning more than under the traditional calendar when schools used to start after Labor Day.
June and I haven't been doing a whole lot right now after coming back from visiting Ralph and Lisa in South Carolina. What a nice visit that was! The StreetJelly concert was a super highlight, but it was good all around.
Speaking of StreetJelly, I'll be doing my "regular" show tonight (Thursday) 9 pm EDT. In fact, that's only about an hour from now, so I better hurry up. Back to just me, Sarah (the guitar), maybe some other instruments, and you, my faithful viewers and remote back-up band. The theme is the generic "Summer Sunshine," and I'm bringing out some songs I haven't done in a while. Please join me and help me have fun!
And furthermore, speaking of playing the guitar, my painful "trigger finger" has returned. It showed up last year somewhere around this time, and at the subsequent doctor's visit, an injection of cortisone did the trick--for about 10 months. So, at a visit yesterday the doctor offered to let me get more shots at decreasing intervals one after the other ... not a good thing ... or get surgery. Actually last year, I had expected surgery, and the good results of the shot were a real surprise, but, obviously, that's not the kind of thing someone would just want to maintain over and over again. So, I'm scheduled to have the problem fixed on September 9, one day after the anniversary of my day of amnesia. It'll take a couple of weeks to get back to normal, and it's supposed to remain that way then. We'll see what happens with my musical efforts during that time.
The summer Olympics are well underway, of course, with all their usual flamboyance, real and contrived drama, and some awe-inspiring accomplishments. It's the one time every four years that we all get interested in gymnastics, and the women's team has definitely made it worthwhile. Congratulations to the "Final Five" for an entertaining gold medal performance.
Well, here's another occurrence of phi in nature: the genealogy of a male bee, commonly called a drone. We get to it by way of the Fibonacci series.
First, some basic facts.
1. Your basic worker bee is a female bee descended from an ovum supplied by the queen bee and fertilized by a drone. She collects the nectar in blossoms to make honey, distributes pollen among flowers in the process, and stings people if they deserve it from her point of view. Worker bees have no role to play in reproduction other than the coronation of a new queen if one is needed.
2. A queen bee is a female bee who has been fed large amounts of "royal jelly," a product that turns her into a reproductive machine. She is the daughter of a previous queen bee and a drone.
3. I've mentioned the drones above as the male bees who fertilize the eggs produced by the queen bee. That's what drones do. In fact, it's the only thing that drones do. The unique feature of a drone is that he does not have father, but only a mother, namely the queen bee. Consequently, he only has one set of chromosomes, as opposite to the normal two. Scientifically, it's called "haploid," a term that refers to having only half of the usual numbers of chromosomes (in contrast to the more usual "diploid").
On the right is a diagram of the parentage of a working bee. Father drone and mother queen give birth to a girl-bee, who will probably never become a queen. If for some reason, she were to be chosen to become the next queen, her sisters and half-sisters will feed her large amount of "royal jelly." Her abdomen would enlarge, and her ovaries would morph from being useless vestiges to high capacity organs. Otherwise, she's condemned to a life of celibacy and hard work. All humor aside, the queen probably works harder than all of the other bees, spending her life giving birth to one larva after another, and it doesn't look like much fun.
So, let's look at the ancestry of a drone. In the diagram below, you can see that every drone had one queen as parent, and every queen had two parents, a drone and another queen. Common female worker bees have no place in this depiction because they do not actually participate in the reproductive process. And, needless to say since you can see it yourself, the number of ancestors of the drone follows the Fibonacci numbers as we go back generation by generation. Thereby, they are headed ultimately to converge at phi.
This entry comes to you from deep in the American South, where we are visiting brother Ralph and his wife Lisa for a few days. This has been a totally spur of the moment trip, the kind of thing you can do once you’re retired and the nest is empty. June is continuing to struggle a bit health-wise, and we’re looking forward to her appointment with Dr. B on August 17. Speaking of that date, you may also recall that it’s the day on which we celebrate the birthdays of William Carey, Davy Crockett, and some other really cool people.
We’re having a great time together here. In case you didn’t see the post on FB, on Thursday night Ralph and I sang and played together again for the first time in quite a few years. There was a time when neither one of us thought we would ever perform again as individuals, let alone as a duo. So, last night's StreetJelly.com was an incredible blessing for both of us. We had a total riot, and I think that many folks in the audience had a good time, too. There’s a video of us rehearsing on my Facebook timeline.
Now I'd like to come back to the topic that I may have treated too cavalierly earlier, namely the appearance of the golden ratio and ϕ in nature.
Phi in Nature
Do I need to define “nature”? I would think not, but it may be helpful if I emphasize a few of its characteristics. Somehow I want to maintain the idea of “nature” as that part of the created order that has not been formed by human beings. Of course human beings are themselves a part of nature in important ways. So, maybe rather than saying “formed,” a better word might be “subdued” or perhaps “governed.” A sunflower is a part of nature; a picture of a sunflower, strictly speaking, is not. Nature includes our bodies (to a large extent anyway) as well animals, plants, germs, diseases, rocks, astronomical objects, subatomic particles, and lots more stuff. Nature manifests signs of being made by an intelligent personal being, though nature seen as a whole is neither personal nor intelligent. It contains personal intelligent beings, but nature per se cannot think, plan, wish, imagine, invent, like or dislike. For example, contrary to the popular saying, nature does not “abhor” a vacuum. The regularities that we see in nature lead us to conclude that a vacuum will be filled with air or some other gas as soon as soon as access is provided. However, nature has no opinion on this issue—nor on any other, for that matter.
A recent article in the Smithsonian (Danny Lewis, “Why the Turtle Grew a Shell—It’s More than Safety”) introduces us to some recently discovered ancient turtle fossils. These newly available specimens shed fresh light on the reason why turtles may have developed shells. It was not for the sake of defense against other animals, as we might think, but as an aid for them to burrow underground in order to survive the inhospitably hot climate of South Africa, their abode at the time. The article closes on a cautious note:
"While more research needs to be done to determine whether the earliest turtles known to have shells were diggers themselves, it just goes to show how adaptable nature can be."
Well, I’m afraid I need to add one more item to my list above: nature cannot be “adaptable” either. The term implies foresight and intentional reactions; but the impersonal forces of nature cannot qualify as being “adaptable” in a meaningful sense. You can say that a certain life form became adapted thanks to the potentials found in nature or that some species have turned out to be more adaptable than others in hindsight. However, any kind of intentionality on the part of “nature” or the species in question is overstepping the boundaries of evolutionary biology—unless you accept that there is a Creator and Sustainer who has been supervising the entire process. If you see something “smart” in nature, it’s not nature itself that’s smart because nature itself has no mind. The smartness points us back to the one who made nature.
Having said that, there are some wonderful examples of apparent smartness in that collection of things in nature, and, unsurprisingly, some of them are classified along that line due to the presence of the golden ratio.
How can we know if some item or phenomenon exhibits ϕ?
1. For some things we can take measurements to see if there are lines that are related to each other in the golden ratio. This is an easy method for human architecture and pieces of art, though we also have a large amount of room for fudging there. It’s also possible to do so for some natural items, such as crystals. For the most part, though, it’s not a practical way to proceed.
2. Bring on the Fibonacci numbers. It appears that some things come only in numbers of the Fibonacci series. Furthermore, if we can find relationships based on the Fibonacci numbers in nature, we can avail ourselves of the fact that they do converge to ϕ, and so the series in question can be said to be in keeping with the golden ratio.
3. Measure the angles. This approach is particularly applicable to anything that appears in spirals. Remember that we said that logarithmic spirals, of which the golden spiral is a special case, distinguish themselves from Archimedean spirals by the fact that their increase in length maintains a consistent ratio, so that the spiral moves away from its point of origin at a much faster rate. In that case, any line drawn from any segment of the origin of the spiral should manifest the same angle.
Thus, in the graph above, the angle at which any of the three lines intersect the spiral is the same. Livio helps us calculate the value of that angle for the golden spiral.
360° ÷ ϕ ≈ 222.5°
Because this result crosses the 180° line, we should come to it from the other direction.
360° - 222.5° = 137.5°
This is the angle at which all the lines cut the spiral whenever they intersect it. If the origin of the spiral converges to the center of the graph (i.e. point 0,0), then the x and y axes will also intersect with the spiral at that angle.
Please keep in mind that a golden spiral is a logarithmic spiral, but not every logarithmic spiral is a golden one. Nevertheless, Livio makes the case that golden spirals are very likely the ones in nature. Apparently, in whatever way living beings expend energy in assuming formations, the golden spiral based on phi is the least demanding. So, when we see a logarithmic spiral in nature, the poor chambered nautilus notwithstanding, there’s a good chance it may also be a golden spiral.
Let me try to explain the above point a little more. Livio characterizes phi as the “most irrational” among the many well-known irrational numbers. Let’s make a quick comparison between ϕ and π. In some ways, ϕ is more approachable than π; in others it isn’t.
Can it be derived from an algebraic formula?
No. π is “transcendental.”
½ (1+√5) = ϕ
Can it be approximated by a fairly simple rational fraction?
Calculating ϕ involves referring to another irrational number, namely the square root of 5. The approximate value of one irrational number is computed by including another irrational number. Consequently, if any items are arranged in a way that makes use of phi, no two items are ever going to be sharing one or the other coordinates in space.
[A similar thing applies to time. On occasion one hears the idea that, given an infinite amount of time, it becomes, not only probable, but even certain that every event of the present or past will recur at some point in the future. But that’s not true, as can easily be demonstrated with the following mind experiment conceived by Georg Simmel (1858-1918) and reported by Walter Kaufmann (1921-1980) in his magisterial book, Nietzsche: Philosopher, Psychologist, Antichrist. (Fourth Edition, Princeton University Press, 1974, 327). It uses pi, but phi would work just as well in light of the above considerations. Think of three disks rotating at different speeds. One completes a full rotation every minute. The second one does so every 2 minutes (thus, running at half the speed). The third one has a cycle of 1/π minutes.
The disks start at a point we can call zero (maybe the twelve on a clock face). That particular arrangement will never occur again. Similar events, though not formally devised by anyone, occur frequently since pi as well as other irrational numbers come up over and over again in everyday life.]
So, let’s look at a particular example now and a few others next time.
Flower petals and plants
1. It is an observable fact that the number of petals on many species of flowers come in Fibonacci numbers. Once again, I have noticed that some of the same pictures appear website to website, and I cannot credit the person or organization that created them in the first place. Here is a photo with a set of pictures that demonstrate the “preference” of many flowers to take the numbers of their petals right out of the Fibonacci series.
2. Even more interesting is the phenomenon of how the petals of many flowers are arranged. Petals and leaves frequently form a logarithmic spiral around their center. Here's why this pattern is a valuable asset for a plant.
Imagine a flower with several rings of petals, and it takes exactly four petals to make up one ring. So, we start out with a nice flower with four petals surrounding the center.
But we stipulated a flower with several rings of petals, and the second ring also follows the rule of four to a ring.
Clearly, this isn’t going to work too well because the second layer will just cover up the first. And if we continue that pattern, the third layer will cover up the first and second.
The flower is in trouble. How might it work better?
The flower would be far more functional if the relationship among its petals was not governed by such an easy integer as four. But, if we stay with straightforward rational pattern, the same thing will happen. Increasing the numbers of leaves by decreasing the ratio with larger integers, we would just get more layers of petals or leaves pancaked together.
We could play out this scenario further, but you already know where I’m heading with this. In many plants, the petals are arranged according to the golden angle of ca. 137.5°, and thus form a golden spiral. Thereby exposure to sun and rain, the attraction of insects for pollination, or whatever else the plant needs to thrive is maximized. Rather than my pitiful drawing above, we can see a beautiful rose that incorporates a rather complex as well as beneficial mathematical pattern in its structure.
Is this amazing? I think so. Does it point to the idea that a very knowledgeable and powerful being must have overseen the creation and further adaptation of flowers? I should think so.
There is a point that someone could make and unreflectively think that it counted against the above idea. It could be, as Livio hypothesizes, that the Fibonacci sequence and formations according to ϕ represent the least expenditure of molecular energy in the formation and continuation of plants. That’s a good idea, and I already mentioned it above, and—in fact—it is a great idea. It so great that we’ve only pushed the issue one step further back. It only increases the wonder of the phenomenon, and, thus, leads us to an even stronger conclusion that the principle was installed by a supreme creator.
More examples next time.
It's cooled off a bit compared to last week. So, I got a little bit of gardening done yesterday. Also, the swimming pool of Smalltown, USA, is open once again. It was closed all last week because it is located right along the fair grounds, and last week was the 4-H county fair. Neither June nor I were particularly interested in the fair this year, partially because of the heat.
I paid a visit to Dr. M, my skin doctor, today. I expected him to scold me for not wearing long sleeves all of the time or not using sun block enough. He would have been only partially right, had he done so. As it was, he skipped that part and gave me a stronger cream to put on my arms. -- No, I'm going to post any pictures dealing with my skin.
Well, I didn’t get much of a response to my golden-spiraled swan, not that I expected to cause a buzz with it. I had two purposes in posting it. One was to see how one can make things look as though they incorporate ϕ, even for a moment, until one takes a closer look. The poor swan in the picture has absolutely nothing to do with the golden ratio, at least to the best of my knowledge and analysis. With a little adjustment of the picture, I got it to the point where I could make the spiral center focus on his head, and then the body fit pretty nicely into the back, actually. The obvious problem is that much of the left side of that picture contains nothing but water. So, the total dimensions for the picture notwithstanding, the swan has his own proportions, which don't come out anywhere near the golden one, once you let the water drain out. The other purpose I will tell you about some other day.
I am working through and improving the entries in this series on its separate site. In the process I am not correcting the previous blog posts, unless I run across something so utterly egregious that I don't ever want posterity to see it. Please use the omnibus site to catch up or refer back to.
I have really honestly wanted to get to the Fibonacci numbers and ϕ in nature, but as I keep sorting through material on the golden ratio on the internet, I keep running across its applications in art and architecture that really need some commentary. Where and how the "golden magic" is applied is almost scary. It seems as though there is nothing that we might consider to be beautiful that has not been given the ϕ treatment by golden-ratio-enthusiasts (hereafter: GREs).
Golden ratio obsession (as opposed to its simple recognition and use) can be traced to to an Italian mathematician, Fra Luca Pacioli (1447-1517). (See Livio, 128-37, for more on this and the next few paragraphs.) Up until then, the ratio had been called by its original name, "the proportion between mean and extreme." Pacioli wrote a 3-volume work on it, entitled, De divina poportione -- On the Divine Proportion. He saw multiple spiritual meanings in it as something that was both uniquely created by God and expressive of God's nature. However, he did not go so far as to see the golden ratio in every nook and cranny.
Pacioli and Leonardo da Vinci were brought together by fate and politics, and Pacioli taught Leonardo about the "divine proportion" insofar as the latter may not have had not had previous knowledge about it. In return, as Livio (133) put it, Pacioli had the "dream illustrator" for De divina poportione since Leonardo provided the pictures of geometric solids and other graphics for the book. Consequently, there is no problem with looking for the golden ratio in some of daVinci's works and being confident about it if one has found it clearly in certain places. Of course, GREs have spared no effort in finding it in all of Leonardo's pictures, multiple times over in certain cases. Sadly, their unsparing vigor winds up concealing genuine substantiations of ϕ underneath the multitude of invented versions.
For example, efforts to include the Mona Lisa in that group suffer from many of the same problems as those we’ve mentioned before with the Parthenon and the Taj Mahal. People appear to begin by assuming that the golden ratio must be there, but then have to find some place where it might actually fit. For this part, I’m going to inscribe my own drawings on the Mona Lisa, based on what I’ve found on the web. That way, I won't pour any more rain on anyone’s parade in particular. Many versions are just copied and pasted from site to site. If you want to see these and others, some of which are just plain bizarre, look for them on Google or your personal search engine of choice.
After looking at numerous contrived examples, it appears to me that Mona Lisa’s face can actually be very nicely circumscribed by a golden rectangle. It does not strike me as an ad hoc imposition.
Fig. 1 Mona Lisa's Face surrounded by a golden rectangle
This picture limits the rectangle to the open face, taking the measurements at the longest and widest extensions. In other words, there is good reason to believe that the dimensions of the rectangle mean something.
One problem is that GREs are not content to find a single likely instance and must build nested rectangles, spiral, triangles, pentagons, pentagrams, and geometric objects Euclid would never have dreamed of that are somehow supposed to contribute to the painting's golden-ness. Obviously (at least to you and me) things don’t work as well with such imaginary placements. Consider the picture below. Again, I have redrawn it based on multiple instances of its appearance on various websites.
Fig. 2 Half of Mona Lisa's head decorated with a golden rectangle
What’s up with that? as the saying goes. This rectangle is a little bit larger as a whole than the previous one; it includes the top of the hair. The proportions are correct, but this rectangle doesn’t frame anything on either side. On the left, it loses itself in the countryside; on the right it cuts through her hair, eye and cheek. The reason it is placed there is because it is actually a part of an assemblage of golden rectangles. Thus, we can create a larger golden rectangle by adding a square to the right of the one that's there, though the resulting rectangle also has no clear moorings on the canvas.
Fig. 3 An extended golden rectangle stuck to Mon Lisa's head
We can go on from there, if we wish, and make more rectangles. My point is once again that, regardless of whether one can decorate the picture with one or more lines of golden proportion, if they don’t have a direct connection to the work of art, there doesn’t seem to be much point to it, and it becomes doubtful that the artist intended to create that particular pattern. Thus, Fig. 1, seems to reveal a golden rectangle. Figs. 2 and 3 strike me as highly implausible.
Speaking of Leonardo and Pacioli, the figure that led to Leonardo's later drawing of the "Vitruvian Man" is also described in De divina proportione, and it would be easy to conclude that, therefore, he, too, must obey golden proportions. That just goes to show how easy it is to conclude something wrong, and for that idea to take on a life of its own under the guidance of the GRE's. There is an excellent treatment of this drawing by Takashi Ida of the Nagoya Institute.
Why is this famous drawing called the “Vitruvian” man? It is based on a description of human anatomy by the first-century Roman architect named Vitruvius. His description of the human person, endorsed by Leonardo, emphasizes rational proportions and symmetry. It is in a section of the book by Pacioli that explores various kinds of proportions and ratios, not just the "divine" one.
Did I just say “symmetry”? What an interesting thought! Could it be that symmetry also arouses a sense of beauty in us? Vitruvius thought so, echoing an idea propagated by Aristotle and held to some extent by most writers until golden ratio fever set in.
Symmetry also is the appropriate harmony arising out of the details of the work itself: the correspondence of each given detail to the form of the design as a whole. As in the human body, from cubit, foot, palm, inch and other small parts come the symmetric quality of eurhythmy. [Vitruvius, On Architecture, Frank Granger, trans. (Cambridge: Harvard University Press, 1970), 26-27; cited in the article "Beauty" in the on-line Stanford Encyclopedia of Philosophy.
St. Thomas Aquinas left it open as to what the right proportion may be in a given instance,
There are three requirements for beauty. Firstly, integrity or perfection—for if something is impaired it is ugly. Then there is due proportion or consonance. And also clarity: whence things that are brightly colored are called beautiful. [Summa Theologica, vol.I, q. 39, a. 8, cited in Stanford Enc.
We cannot occupy ourselves with such alternatives for the moment because I must press on.
As far as I can make out, whereas Leonardo golden ratio enthusiasm is a long-standing phenomenon, it is only in its early stages for Michelangelo. Apparently it received its big impetus due to a line discovered on the picture of the creation of Adam in the Sistine Chapel ceiling. (Thanks to David O. for calling my attention to this last year.) It has been discovered that a straight line between the edges of the ceiling segment passing through the point where God's and Adam's fingers almost touch turns out to be in the golden ratio, with the dividing point located right between the fingers.
I have no quibble with this discovery. Of course, the prattle along the line that "now we know why we have always liked that picture because the golden ratio was there even though we didn't know it," is silly. I can't imagine that the picture would be any less appealing if it were shorter by a few inches on one side. For that matter, if I'm not mistaken, there are plenty of compartments of the same length where you can't plausibly stick a golden ratio. So, even though they do not carry the same significance as the one with God and Adam, would I really want to say that they are of lesser beauty?
Next time I really hope for sure: ϕ in nature.
Here is a picture of a swan, and I've framed it with the golden rectangle and the golden spiral nested within it. Is this a good example of the golden ratio?
That's all for now. Please let me know what you think, here or on Facebook.
This entry is a continuation of the previous one, as well as being a part of the lengthy series of phi. I might just mention that, in terms of our physical states, June and I are doing okay. We are still waiting to hear what Dr. B is going to do about some of June’s test results of a couple of weeks ago, and waiting gets old pretty quickly, as we all know.
So, once again I escape into the real world of real numbers, exploring the uses and misuses of phi. I mean, what we used to consider the “real” world has become so surreal at the moment that I can’t bring myself to write on it. Besides, there’s no shortage of commentaries on that realm. So I seek shelter in the part of the world that’s not going to change and its Creator. If you’re tired of reading about phi, or never were interested to begin with, I understand. But please don’t repeat the time-worn myths after passing up this opportunity to reflect on the matter under the gentle guidance of your devoted bloggist.
In the previous entry, I focused on one particular set of pictures supposedly showing how the golden ratio shows up in the entryway to the Taj Mahal, but that this case rests on a glaring mistake, which anyone should have been able to catch apart from knowing any math. Actually, if you examine a larger area than the entry gate, the reason for this strange placement of the golden rectangle entry becomes a little clearer, but no less arbitrary. There seems to be an additional desire for bigger and better golden rectangles, and the inconsistency concerning the entryway not only remains, but is actually expanded by some further dubious interpolations.
I’m going back to the broadly circulated set of pictures that I labeled Pictures 1 and 2 last time. Here is Picture 1 by itself, and, in order to make it easier to talk about what’s happening here, I’ve labeled some important junctures of the lines with letters.
Among the various lines, two apparent golden rectangles are created. One can be described by ACFH, and the other one by BDEG.
These two rectangles can only be golden if they overlap. My measurements fall into the levels of tolerance that we cannot help but allow for. Each rectangle starts from the inside of the opposite decorative door frame and goes either left or right to the edge of the building as it is visible in a picture straight onto the front. These extensions are supposed to be squares, and we know that a square added to a golden rectangle creates a new, larger golden rectangle.
In this illustration, the two squares are ABGH and CDEF—together with rectangle BCFG—are thought to make up two new golden rectangles.
The idea of two golden rectangles created by overlap is clever, though it would have required a lot of subtlety on the part of the architects of the Taj Mahal. I can’t say how such a construction would fit in with the supposed aesthetic appeal of phi. Is our vision supposed to shift back and forth, first catching this rectangle, then that one? For all that I know, such may be the theory and, given the initial assumptions, it could be true, I suppose. But it’s also a leap, and I'm not convinced of the assumptions. We’ve already found that the golden rectangle, as placed in in the entryway, compromises the light-colored decorative frame as a feature of the building (which is then covered up in the alleged close-up shot that I have called Picture 2 in the previous entry). I don’t know which idea came first, the two larger, overlapping golden rectangles or the imposition of the golden ratio on the entryway. Regardless, in either case, the arbitrary choice with regard to the doorway is still a hindrance.
Moreover, we also need to question the geometrical integrity of the two supposed squares. It should be immediately obvious that points A and D have no architectural anchorage whatsoever. They appear above the balustrade cutting through the small turrets at no particular locations of interest, except that they mark the end of the straight line outward from C to A and from B to D. The internet illustration make it just as clear as my depictions. Here is how it looks on one side:
The fact of the matter is that these squares simply do not exist. The front of the building ends before the square is finished and the wall is bent at an angle to form a new facet of the chamfered corner. (I just learned the word “chamfer.” It’s a decoration on what would otherwise be the stark edge of a 90° corner.) In the picture below you see how the top line of the supposed square ADFH does not stay on course heading left from B to I, though one can blame the photographic angle for that apparent anomaly. However, the shift at points I and J is integral to the building itself since the building has an edge there and begins a new facet. The decorations of the Taj include some optical illusions, but this is not one of them. There is no square here, but an edge in three-dimensional space, and I don’t think that it would occur to anyone looking at the Taj in real life, rather than as a flat picture, to see a square at these locations. I, for one, didn’t. Consequently, when combined with the contrived rectangle of the entry way, there is still no golden rectangle here. But, as I keep insisting, numbers are beautiful, but beauty does not depend on certain numbers.
I promised a couple of entries ago that, if I were to run across the misleading information concerning the Taj Mahal and the golden ratio again, I would point out the website that carried it. As it has turned out, after a little more searching I found that the picture I had in mind is not on just one site, but is being copied from site to site, apparently without anyone looking at it too closely. In fact, it is sometimes paired with a second picture that is clearly inconsistent with the first one. The site I chose to mention comes from "Project Steam," and it is written in a gentle and friendly manner. The author provides a catchy response to the idea that, since phi's decimals extend to infinity, it cannot be applied anywhere as a measurement. If I may quote,
Now, I am not a mathematician; I am an artist. So my reaction to this argument is to shrug and say, “Meh, close enough.” If that sort of blasé attitude offends your mathy sensibilities, you should probably stop reading now.
The math fan inside of me is taken aback; my writer's instinct loves the prose. I compromise and say that, if you're writing about a subject, you still should avoid obvious goofs. I don't see any math errors. However my concern should be as clearly visible to the artist as to the recreational math fan, maybe even more so. As you know, I have been measuring the purported golden rectangles as depicted in various images by pixels. It's super-easy to do so with Paintshop Pro, and I assume similar programs, such as Photoshop, are just as good in that respect. However, in this case, measurements are totally unnecessary. You can see the manipulations without knowing any math. I doubt that the author of the post is the creator of these picture, but still, I don't understand how he could post them without noticing the glitch.
Here are the two pictures in question, already combined into one image on that website so that a discrepancy should be easy to catch.
Picture 1 (PS) Picture 2 (PS)
Picture 3 (mine) Picture 4 (mine)
I had remembered the adjustment correctly when I reported on it. Specifically, I recalled the placement of the golden rectangle at the entrance (Picture 1 and reproduced from memory in Picture 3), which did, indeed, come close enough to the value of phi to satisfy not merely the artist, but the casual math player in me. But then I realized that there was an issue with how the rectangle was arranged, namely with the top being above the light-colored frame, and the sides inside of it, as highlighted with the teal lines in Picture 4.
And now, if I may, I would like to direct your attention to the close-up as shown in Picture 2 on the top right. In that picture the rectangle encloses the entire outside of that decorative frame. The difference is visually undeniable. My measurements came out to a ratio of 1 : 1.3, not close enough for anyone I would hope. How can one miss the difference in where the lines are drawn? Maybe the artistic author re-posted the pictures as he found them wherever he found them and didn't pay much attention because he didn't think it would make any difference. But it does; phi is all about a number, and a different number can't substitute for phi. I don't intend to drag out a lecture on the virtue of precision in whatever you do. "Good enough" may be--no, it is--good enough at times. However, if you're illustrating the golden ratio, and the illustrations are not illustrating the golden ratio, what good is what you're doing? The choice here seems to be between either an arbitrary golden rectangle that disregards the architecture and decorations or a rectangle that follows the features of the building, but not the golden ratio.
My point, once again, is simply that golden-ratio-mania is leading people to find phi all over the world, linking the beauty of a building to a specific number, and imposing it on objects where it is neither present nor needed. I'm not opposed to finding the golden ratio where it is, and if it's contributing to the beauty of something, very well. But if we feel as though we need to find phi in some contorted way in every beautiful structure, we are doing ourselves and the object a disfavor because then we are mechanizing our aesthetic sensibility.
I have had the privilege to visit the Taj Mahal. When you first walk onto the compound you cannot see it because your view is obstructed by a rather high wall, and you're too close to look over it. Then, once you walk through the inner gate, there it is, the Taj Mahal, right in front of you, larger than life--and incredibly beautiful. No reproduction that you've seen before can really do justice to the magnificent structure now in full view. Does it embody the golden ratio? I can't find it. Is it an unbelievably gorgeous sight? Absolutely. Would it be more beautiful if it did manifest the golden ratio? I don't see how it could be.
The time here in Mirkwood* is slowly coming to an end. It's been a good stay. The weather has been extremely cooperative.
(*I assume you realize that I'm not using real place names.)
The restaurant here attached to the "Prancing Pony" in Bree has gone bonkers. It used to be a basic country-style place where you could get a good meal for a decent price. Now they're charging $10 dollars for just a cheese burger. That's like Hilton Hotel prices, and not exactly in line with out budget. So we take food from the local grocery store to our room. But that's not what I wanted to tell you about.
The first thing I've done every morning has been to go to the stable for my day's ride, while June slept in. I've mentioned the names of some of the horses I've ridden in this park: Whiskey, Bailey, Dan, Levy, Chip, and Bonnie are ones that come to mind. I rode Bonnie once a year ago, and she was my first horse on Monday of this week. As you can imagine if you know me, I always try to strike up a friendship with whatever horse I'm on, talk to him or her, and treat them to some of my cowboy songs, hoping I'm not annoying whoever is riding right before or behind me. So, today I was sitting on a bench by the horses' enclosure, just a board nailed to the fence, with my back to the equines, watching the world as I was waiting for a few more participants to show up for the ride. All of a sudden, I felt a horse nestling the back of my neck. I turned around, and there was Bonnie, saying hello to me. We chatted for a few moments, and then she ambled over for her breakfast hay. I mentioned that little serendipity to Sierra, one of the trail guides, and I added that I did not fantasize that Bonnie really remembered me. But Sierra said she very well might have. Either way, it was nice to have a horse be nice to me.
And now back to the miracle of phi and the golden ratio.
As I said in the last entry, we need to come back to the appearance of phi in the natural realm. Earlier on I skipped most of it, except to give a mere mention of a few examples and a couple of websites. However, I didn't go any further into depth with it, eager to do more math and philosophy than to catalog the appearance of the Fibonacci numbers in nature. That's been done and overdone, I mused. However, on second thought, to do justice to some of those occurrences, we can steep ourselves a little bit more in conceptual matters, and so I'm returning to it. This material may eventually wind up towards the middle of the completed website, right along the other relevant material. One of the first things I've done is to get rid of my earlier rather awkward animation of a golden spiral and to substitute one from Wolfram Alpha, which I then subsequently animated.
I'll reproduce it here so that you don't have to shuttle back and forth:
There are two important groups of spirals, mathematically speaking, and most spirals that we encounter fall into one or the other group: Archimedean spirals or logarithmic spirals. The former are named after Archimedes of Syracuse (287-212 BC), and one member of this set bears his name, "Archimedes' Spiral." Archimedes was frequently engaged in applied math (e.g. war catapults, discovering the principle of density displacement), but also made some important contributions to the more theoretical side of math. He came up with a break-through method of calculating the value of pi. As I'm thinking of him and his work on spirals, I cannot help but think of the occasion of his death when he was killed by a Roman soldier. According to a popular, albeit unreliable story, when the soldier entered his room, he was engaged in studying some geometric figures, and he was supposed to have said, "Do not disturb my circles!" Was he maybe making further advances in his study of spirals?
Logarithmic spirals are associated with Jakob Bernoulli (1655-1705), who makes another appearance in this series in connection with the "Basel Problem." Bernoulli was so impressed by this kind of spiral that he called it the "miracle spiracle" spira mirabilis, and asked for it to be a decoration on his tombstone. The tombstone artisan clearly had not studied up on the nature of the spiral in question or perhaps did not understand it. He made a "plain old" Archimedian spiral instead. I don't think that Bernoulli cared any longer at that point, but the difference was very important to him during his life time because it illustrated for him the way in which a thing can be changed and yet remain the same. Specifically, he saw in the logarithmic spiral the renewal of the person entering eternity in heaven. And actually, if it hadn't been for that mistake, we probably would not be talking about it as much.
Just a day or two ago I saw on an apologetics website a notice of a new book, which supports the claim that evidence for God can be found throughout nature. I don't remember author or title, and I don't want to embarass him or his publisher or me in case I'm going to be wrong on this point. All I know about this book at the moment is what the notice said and the picture on the cover, which includes an Archimedian spiral. Books of this nature usually include discussions of the Fibonacci numbers and the various golden angles, rectangles, triangles, and spirals. So, as I glanced at the picture, I remembered Han Solo's words, "I have a funny feeling about this."
So, what's the difference? I'll put it in terms that I can understand and spare us the formulas.
1. Archimedean spirals. Imagine a garden hose that lies flat on the ground, neatly arranged in ever-expanding circles. The circumference grows with each rotation, but the distance between each individual piece of the hose and its adjoining ones remains the same. The radius from the center increases, and so does length of each arc. Consequently the angle of expansion (slope, derivative) of the arc flattens. Here is a picture of the Archimedes spiral.
2. Logarithmic spirals. Take the same garden hose. Start to coil it up a little. Measure the relationship between the radius and the arc. Now coil the hose a little more and maintain the same ratio. Well, the segment increases, but you're maintaining the same ratio and thus the same angle of expansion (slope, derivative).Then the distance between segments must increase concomitantly. Extend the coil some more. Again, keep the same ratio, but extend the length of hose. Your distance between hose segments will be increasing some more. As you keep going in that fashion, the spiral becomes looser and looser, and that's because it keeps its ratio. If you were by chance to see just one segment, and you had no idea of how large the magnification was, you would not be able to tell the position of that segment relative to the point of origin of the spiral. Here is a picture that's not in perpetual motion.
A logarithm is the flip side of an exponential number. Express the number 100 in exponential terms, using 10 as your base. You will write 102.
Now you can say that 2 is "the logarithm of 100 to the base 10."
Logarithms are helpful in many ways. They decrease the distance between numbers to a manageable size.
Expressed exponentially 1,000 is 103, and so its log to the base 10 is 3. A difference of 900 is expressed with just one integer, going from 2 to 3.
10,000 is 104, and this time it's a gap of 9,000 units that's expressed with an increase of yet a single number, namely its log 4 (base 10).
The pattern goes on in the same way. Furthermore, if you need to multiply two numbers, such as 100 x 10,000, you can take their logs, 2 and 4 respectively, and just add them. Then you can go back to the exponential version and write out the result as 106, which is a million. Obviously you don't need logarithms for making simple multiplications or divisions for the powers of 10. However, all numbers have a base 10 logarithm, and so a table of logs can help you with slightly more complicated procedures.
Some of us who have lived through more history than other readers may remember the good old slide rule, which was eventually replaced by the pocket calculator, though not for a few years after I had finished my undergrad as a science major (zoology). It was extremely useful and amazingly accurate. I won't go into its design or functionality now, except to show how the display on a number line (and it had several) was based on a logarithmic scale. You've barely begun with the number 3 by the time you get halfway, but the the distance for each number goes down on a logarithmic function, and you get all of the first ten numbers on each bar.
Actually, I need to tell you that more often than not, it's not quite that simple. Mathematicians and scientists usually prefer to work with a base different from 10, namely a number that goes by the name of e, called such in honor of the magnificent Leonhard Euler, who is considered the greatest mathematician ever by some people. e is also an irrational number, its value is approximately 2.71828..., and I'm not going into its properties any further. I will just say that e holds as many surprises as phi and pi.
The ratio of increase in a logarithmic spiral can vary from spiral to spiral. As we observed already, the chambered nautilus grows according to a logarithmic pattern, but it's not phi. However, we can find logarithmic spirals in many other parts of nature, where they follow the Fibonacci numbers, and, thus the golden ratio and phi.
It's gotten way too late again. More next time
Thanks to everyone who gave me input on the Golden Ratio test. Here is a table of the results as of 11 pm tonight. Voting is now closed.
What does this mean? Not a whole lot. This was anything but a scientific survey.
Nevertheless. taking it for what it is, we do see that, among those who took the trouble to respond, there wasn't any mysterious attraction to the in the golden ratio. Given these reflections, and, if this quiz has any validity at all, we see that we like the proportion in the range of 1.6, but beyond that , it would be really hasty to draw any further conclusion.
"Can you handle the truth?" as somebody said to somebody else in some movie, whose title I forgot. Of the pictures in the chart, the one that is closest to the golden ratio is number 4. The distinctions have to be quite precise since that's what phi is all about. I must confess that if I had been asked which format I considered most attractive, I probably would have selected no. 3.
Also, on a different note, Thanks to everyone who took part in my StreetJelly show tonight. I never take anyone's attendance for granted, and I appreciate you presence. And thanks for the tokens or pins that you send my way Good night everyone!
This blog entry is coming to you from our little hide-out south of Smalltown, USA. It's been a good two days so far. Originally, we thought of driving further, but at this time that doesn't make much sense. I get to do two of my favorite things every day: riding horses and swimming. Today I got to ride on Whiskey again, my favorite, as some readers may remember. June has very little energy, and we'll get into that some later time. Right now there are so many folks hurting and in urgent need of prayer, so we'll wait our turn. Does that make sense?
Before getting back to phi and the golden ratio, just a word in general in the context of Christian apologetics. There's a line of argumentation that we all would do well to avoid. I've seen it used by Christians in conversation with people of other religions, and I've just now experienced it myself. The argument runs this way:
Person A: My religion/world view is X.
Person B: If your religion is X, then you also believe Y, and Y is absurd.
Person A: But I don't believe Y.
Person B: But you must believe Y if you're X, and so, either you don't know your own religion, or you hold to an absurd belief.
Person A: But I really don't believe in Y and it doesn't fit with what else I actually believe.
Person B: Too bad. I know that you are obligated to believe Y, and you're a fool for believing such nonsense.
Person A (slaps himself on forehead): Okay, I see. You're right. How could I not accept the obvious nonsense that you say is a necessary a part of my worldview/religion? Thank you for showing me my duty to believe something absurd so that you can take pot shots at me.
Maybe Person A is inconsistent; maybe Person B is uninformed. Either way, it doesn't make much sense to criticize a person for a belief that he or she doesn't hold. This dialogue would be funny if it came from Abbot and Costello. But in the real world, it's not exactly the best approach.
Typical examples from Christians:
Jews are obligated to believe the Old Testament and are required to expect the re-institution of temple sacrifices.
Hindus and Buddhists must be pantheists if they only understood their religion completely.
From a non-Christian to me:
As an evangelical Christian, I must accept a young earth theory of origins, which I don't. But it doesn't matter that I say I don't; if I were consistent with my belief in the inspiration and inerrancy of the Bible (as defined in a bizarre way that I hadn't I heard of in a long time, if ever), I would believe in the young earth, and, thus, I'm an idiot for believing it.
This foolishness has to stop.
And now back to our contemplation on the golden ratio and phi. The general belief is that the golden ratio in some inexplicable way catalyzes our appreciation of beauty. Certain experiments have supposedly shown that this idea cannot be documented with evidence. I suspect that this matter is similar to a court case where each side brings in their expert witnesses to support their case. We're talking about a very narrow window. 1 : 1 1/2 is too little; 1 : 1 2/3 is too much. So personally, I would be surprised if eventually there were some clear and objective proof for the aesthetic appeal of phi in art, but it's not something that I can be or want to be dogmatic about. The one thing I do want to caution us about is that, as I have insisted all along, there is a lot of beauty in phi, but its appeal, if any, is not due to some supernatural numerological power .
The table below the video contains a picture of a "Cherokee maiden" that I took ...
[Queue up Bob Wills, Merle Haggard, or Asleep at the Wheel!]
... a couple of years ago. Each one of these pictures has a different ratio of its sides; one of them is in the golden ratio. Is there one that strikes you as more beautiful than others? I sure hope so because some of them are pretty badly distorted. Which one do you like best? I shall disclose which one is in the golden ratio at some future time. In the meantime, have fun with it, assuming that you find this kind of thing amusing or interesting.
We're almost done with this series. Next time or so, I'll bolster the section on the Fibonacci numbers in nature.
The Golden Ratio in Ancient Architecture
It’s time to come back to the point of this series. Its topic is the beauty of the number that is usually called phi (ϕ). Given all of my stacking and popping, I think a little recap is in order:
Contrary to the things you frequently may read, phi is not derived from the series of Fibonacci numbers, though they do converge to the value of ϕ: 1.61803…. Its origin lies in the geometry of a pentagon from which we can derive a “golden triangle,” which is distinguished by the fact that the ratio of one side to its base, ...
... is equal to the ratio of the side and base combined to a side to the base.
If we want to express this relationship with regard to a straight line, we can say that there is a line, connecting points A and C, running through point B,
and the ratio of BC to AB is the same as the ratio of AC to B, namely the famous phi: 1.61803 ….
I recounted some fascinating properties of phi, and showed you a few interesting features of phi in relationship to the Fibonacci series. After an excursion on some wonders in π, I asserted that, for Christians, the presence of the Fibonacci numbers in the universe and the beauty within the world of numbers itself should lead us, without hesitation, to affirm the wonderful hand of God displayed in his creation. Then I set out on a long excursion on some scientists and mathematicians who see the world of numbers as divine, but, not wanting to acknowledge the God of the Bible, find God in some unexplained and inexplicable manner within his creation itself.
Given the length and seriousness that this series has taken on, I will flesh out the earlier section on the Fibonacci series in nature. For now, I would like to address two questions that seem to go hand in hand.
We find beauty in many work of art. Some works may be more beautiful than others. Many people say that the perception of beauty may in certain cases be due to the fact that artists have incorporated the golden ratio in their creations. Thus we have two considerations to address:
1. Do we actually find the golden ratio in some of the works that are usually cited as examples of manifesting ϕ?
2. Does the presence of the golden ratio actually trigger our response to consider some things as beautiful?
We need to address the first question first because that's inherent in being the first question.
Artists are free to incorporate the golden ratio in their works to their hearts’ content. And if we find beauty in their production, so much the better. A good example is Salvador Dali’s painting, “The Sacrament of the Last Supper.” Its proportions in internet reproductions appear to be pretty close to ϕ, and I can’t be sure how much may have been lost in either trimming or framing. In this case, a tolerance of a couple of millimeters or pixels can be taken allowed.
For the reproduction I have picked out of the many on the web, this one comes out at 1 : 1.58. Dali leaves no doubt about his intentions, seeing that he inscribed his painting with a dodecahedron, whose twelve sides consist of pentagons.
Apparently the two most frequently used illustrations of the golden ratio are the chambered nautilus and the Parthenon, the ancient Greek temple devoted to the virgin Athena, located on the Acropolis. We already mentioned that the nautilus increases his cells in the shape of a logarithmic spiral, though the ratio is not ϕ. Nevertheless, it is frequently used. Even the front cover of Livio’s book, in which he clearly states that the nautilus is not an example of the golden ratio, greets us with a representation of this misunderstood mollusk. You see that authors are frequently less in control of their books than one might imagine.
The Parthenon is often used to illustrates how an architect employed the golden ratio to endow his work with beauty. It appears to be almost a given that you can find the golden rectangle in the Parthenon. But we must ask, where exactly in the Parthenon do we find the golden ratio? People making this claim usually illustrate it, and the lines that are drawn differ from person to person. Precision is important. When it comes to pictures of buildings, we’re not dealing with millimeters, but with much larger entities.
Here's one example from the webpage culturacolectiva.com. It finds not just one, but two golden rectangles in the building, situated next to each other, inscribed by a golden spiral. Note how the smaller one on the left goes down to an arbitrary line on the ground.
We can contrast that depiction with what we see on the site Design by Day™ Everything you need to know about the Golden Ratio in Graphic and Design.
As most of these pictures do, it extrapolates to the top of the pointed façade, which is lost. The bottom goes to somewhere in the rubble at the bottom of the stairs. The two sides drop in alignment with the eaves and connect to the base in no-where’s land, cutting the platform where the rectangle requires it, but where there is no architectural indication for it.
Similarly, a site on Greek and Roman art, seems to give priority to the rectangle at the expense of architectural detail.
The location of points A and C in this picture are crucial to finding an additional golden triangle, but don’t seem to play a role in the actual construction of the building. Here is another picture on the same site. Note the pronounced leftward shift of the rectangle.
On Pinterest I ran across this little gem:
The angle at the top is a little more obtuse than in some of the other pictures that extended the lines, and so the rectangle does not need to be as tall to meet the proportions. The base line is located above the stairs. The line is straight, but the placement of the columns is not, except for the front two. And again, the location of the two bottom corners is established by the geometry of the golden rectangle, not by any mark in the building.
We can find the following entry in the Nexus Network JournalTM:
If I read the context correctly, this diagram is intended to make the point that there is no single way of inscribing the golden rectangle in the Parthenon. And that observation obviously negates the idea that phi is incorporated into the Parthenon, and that we see its beauty because of the golden ratio.
Finally, Gary Meisner, on his website ϕ = Phi ≈ 1.61803 comes up with yet another placement of the rectangle, cutting off the eaves.
But he expresses some hesitancy about imposing the golden ratio on the temple front as a whole, because it seems to require too many arbitrary decisions. He then seeks for it in parts of the structure, but, even if that should work, it's out of keeping with the conventional belief that the entire facade attracts us with the golden proportion, and so we'll skip that exercise.
I think that my point should be pretty obvious by now: Lots of people agree that the Parthenon receives its beauty at least partially from the golden rectangle. But there is no unanimous agreement where precisely it is located, a fact that at a minimum should make us a little doubtful concerning this idea. I don't think that beauty is entirely in the eye of the beholder, but the golden rectangle might be.
Actually, a couple of weeks ago, when I was thinking about this topic as a future entry, I decided to try my hand at the process without knowing what to expect. I knew what would be coming with the Parthenon, but I could not recollect ever seeing any pictures of the Taj Mahal in connection to the golden ratio. There are many of them; I had just not seen them. Well, I have a few good pictures of the Taj Mahal that I took in 2006, so I started to select various rectangles on the building and measured their lines in pixels to see if I could get close to a proportion of phi anywhere. The obvious place to start was with the entryway, trying to be as careful as I could, but also being open to the possibility that I could make an error in placement. Nothing I tried there worked out, and neither did the window-like openings.
I finally decided that I could not impose the golden rectangle anywhere on the Taj Mahal without fudging.
Then, in the course of looking for good pictures of the Parthenon, I came across my first Taj Mahal/Golden Ration depiction. I did not write down the URL at the time (or I misplaced it), but I remember exactly what it looked like, particularly because it was indicated in exactly the area I had been searching. My measurements confirmed that here was, indeed, a genuine golden rectangle--or so it seemed.
But then a closer look revealed that the person who posted that picture had been creative in placing the rectangle without taking the decorations into account. His or her top line is clearly above the somewhat lighter frame, but then the sides run along its inside. I thinned out the line a little bit compared to what I originally posted last night and highlighted the frame in order to make the obvious fudge more clearer for anyone who might have not seen it. I think I'm just as happy I couldn't find my way back to that particular URL because that person might just be angry with me. Some people ascribe mystical, supernatural powers to phi and the golden ratio, and probably aren't happy when an obvious fudge comes to light. (If I do find it, I will post it, so you don't have to take my word for it. )
The story of phi and the Taj Mahal resembles that of the Parthenon. "Everybody" insists that it's there. "Nobody" can agree where exactly it is, and there is a lot of fudging going on. I'm not saying that there are no instances of finding the golden ratio in art. But I'm skeptical about how strongly it is represented in some of the traditional supposed examples. I have maintained that there is beauty in numbers; I'm not so sure that works of art depend on numerical values for their beauty. More on that next time.
"Agnosticism" was a word invented by T. H. Huxley (1825-95) to describe his attitude that he did not know whether there is a God. He took the label of the ancient movement called Gnosticism, whose members prided themselves on their esoteric knowledge, and put an a in front of it, taking pride in what he claimed he didn't know. It was also Albert Einstein's favorite label for his religious outlook, as much as I can make out.
I'm going to repeat a few thoughts here that I brought up in No Doubt About It, 88-89. There are two (or more) ways of construing the term "agnostic": constructive and destructive.
An atheist who reviewed the book a number of years ago really took offense at my using the terms "benign" and "malignant." I hope that by now he has gotten over it, though the words are not inappropriate. Bottom line: the constructive form of agnosticism can lead to a growth in faith; the destructive one is irrational and can lead to a person's spiritual death.
Constructive agnosticism can be summarized with the statement, "I don't know if God exists." Well, if you don't, then you don't. For many people coming to this realization is the best thing that ever happened to them. Someone may recognize that this issue is a whole lot more important than, say, whether Pluto is a planet, and start to search for an answer. That's a good thing. For some young people who have grown up within a Christian environment, it can even be a necessary step to make the faith with which they have grown up their own.
However, frequently people use the language of "I don't know" when they really mean "you can't know" or "nobody can know." At that point the agnosticism has become destructive and does not really differ from atheism for all practical purposes. Someone claiming this view must assume that all possible ways of knowing about God have been tried and failed, and, furthermore, that he or she has personal knowledge of all of these tests and their outcomes. This is, of course, not possible. That's why I said that destructive agnosticism is irrational. It presumes an omniscience, which has not been granted to any human being.
Here's the third YouTube video, this one is about Albert Einstein's religion.
There also is a separate Wikipedia article on the religious views of Albert Einstein. It does, indeed, take a lengthy article to compile all the relevant information, and I'm just going to give you a few highlights. The video, some internet sites, and biographical accounts in books are my main sources. I'm just using Einstein as an example and not pretending to put forward a full biography.
1. Einstein called himself an agnostic. I appears to me that many times his use of the term was a genuine expression of humility. He never failed to acknowledge his limits and the limits on knowledge that all human beings share--even when he was simultaneously trespassing them.
2. This humility is one reason why Einstein eschewed atheism and did not have kind words for those who promoted that position. He saw atheism as a destructive world view that robbed people of the transcendence that human beings need.
3. Einstein definitely did not believe in a personal God, including what he thought was the God of the Bible. There was no shortage of people who held that fact against him, as though he had a greater obligation to believe in God than other people. Incredibly, there were prominent Christians who exhorted him that, since he was Jewish, he was giving Judaism a bad name by not accepting the God of the Old Testament. But Einstein, much like Kaku now, was not open to a "God of Intervention." He considered the idea of a personal God, as found in Judaism and Christianity, a childish fantasy used to instill fear in people so as to make them behave. Einstein had some Christian teachings as a part of his early education, and, for all that I know, his teachers at that time may have been using God as a bogey man figure to frighten children into being good. Needless to say (I hope), we're once again looking at a caricature of the biblical God, and--as far as I can tell with my very limited research--he never pursued educating himself further on a more mature understanding of God. Once he was done with it, all that was left was a patronizing smile from his allegedly more rational point of view.
4. Apparently there were several occasions when some media outlet declared that Einstein believed in a personal God, and these reports made him furious.
5. As mentioned in the last entry, Einstein frequently averred a particular preference for the pantheistic God, as described by Spinoza. Let me just reiterate that his admiration of Spinoza's view requires the romantic glasses through which Spinoza was being read starting in the early 19th century. The things that intrigued Einstein about the universe and the possibility of some kind of a deity are the very things that Spinoza reduced to a flat, undifferentiated monism. To be sure, Spinoza's God was rational and orderly, but the beauty and elegance that mesmerized Einstein are not found in Spinoza's Ethics.
6. Just as we saw with Kaku, it is a whole lot easier to compile what Einstein did not believe than what he did believe. As soon as we try to look more closely at the positive side, we see expectations, beginnings, and inconsistencies without resolution. Here are two quotes that I took out of the Wikipedia article. This is the first one:
"God is a mystery. But a comprehensible mystery. I have nothing but awe when I observe the laws of nature. There are not laws without a lawgiver, but how does this lawgiver look? Certainly not like a man magnified."
Einstein is in awe of the laws of nature, an attitude we should applaud. He goes one step further and asserts that these are laws that must have been legislated by a lawgiver. Once again, we're running up against a highly unusual understanding of the laws of nature, at least as it shows up in the phrasing. The laws of nature are not commandments given by God in the way a government makes laws for its citizens. To repeat something I said earlier, the laws of nature are descriptions and perhaps statistical generalizations. Some of them appear to us to be ironclad and unrevisable. They are discovered by people working in the sciences. So, the idea of a prescriptive divine Lawmaker is somewhat odd for the laws of nature, and it was probably not what Einstein was really saying. I think we can agree that what Einstein meant was that the law of nature are such that one is driven to see an intentionality underneath them. But please note also that Einstein overstepped his professed agnosticism in this statement. He ruled out any anthropomorphic understanding of God, but then he must know something about the one who is beyond our knowledge. If he is truly in the dark about God, then he is not in a position to set up rules as to God's true nature.
This point is important because the descriptions of God that Einstein called "anthropomorphisms" are the attributes of God that lead us to understand him as a personal being. From time to time Einstein used anthropomorphic language about God as well.
Der liebe Gott würfelt nicht.
"God does not play with dice."
Does that assertion mean that Einstein pictured God as a person with hands who performs various actions, but refuses to roll a pair of dice. Of course not. Einstein was using an anthropomorphic image to make a greater point about a being that does not literally have hands or could be tempted to entertain himself with dice. And that is the proper method for understanding the Christian notion of a personal God as well. What's sauce for the goose is sauce for the gander. Our language is limited by our earthly environment, and we cannot express what God has revealed to us about himself without using terms from a finite context and applying them to the infinite. (Some of the best philosophers of religion place the language of religion under the technical label of "analogy.") No, God is not a magnified human being, as the atheist Ludwig Feuerbach proclaimed in his Essence of Christianity. But the only way in which we can talk about God is with human language. The alternatives are either to say nothing at all or to utter something meaningless. Einstein did not choose the former option, and his dictates about the nature of God seem short on meaning. One simply cannot declare the ineffable, let alone make up rules for it.
7. We see the same confusion in the second quotation:
A knowledge of the existence of something we cannot penetrate, of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms — it is this knowledge and this emotion that constitute the truly religious attitude; in this sense, and in this alone, I am a deeply religious man.
I am reminded of the controversy of thirty or so years ago with regard to the work of art produced by Andres Serrano. I'm not going to describe it here because the details don't matter. Suffice it to say, it was rightly considered blasphemous and offensive by Christians, particularly Roman Catholics. However, I ran across one review [Art News 89 (April 1990):163], in which the author defended Serrano. Specifically he said that critics were misunderstanding him; he was not an opponent of religion. In fact, he was a deeply spiritual man who was devoted to a religion focused on bodily fluids. That case certainly is crassly different from Einstein and his beliefs, but the same logic appears to be at work. Whatever Einstein declares to be a religion, regardless of whether it violates any reasonable understanding of religion, is a religion.
religion ≡ whatever Einstein declares to be a religion (def)
Einstein confesses to the existence of something that is almost entirely inconceivable and ineffable. That recognition makes him a deeply religious man, he claims. Once more we are allowed only a scanty look, and the picture can't be filled in because there isn't even a sufficient outline. All we have is a vague apperception that overwhelms us as we study the cosmos. If that's sufficient to be called a religion, then Einstein was, indeed, a deeply religious person. I must admit, however, that I think we're looking at something extremely thin here.
So what? someone may respond. If Einstein was content with that description as his "religion," why is that my concern? Why am I picking on Einstein (and on Michio Kaku earlier)?
If you are asking that question, please read that quotation again and take cognizance of the claim that he is describing the "truly religious attitude." Albert Einstein is not just asserting that this recognition of something beyond is a religion in its own right; he is making an exclusive claim for its truth and superiority. Consequently, that quote does challenge us to interact with it. I cannot help but see Einstein asking the world to emulate his own highly underdeveloped notion of spirituality, inspired by reason but beyond rationality.
I'm going to add one more statement that is going to come across as pretty harsh, I'm afraid. One could say that, even taking all of the above into account, we should pay greater attention to what Einstein is saying in the realm of religion and philosophy because he was, after all, such a wise person. At least some of my present readers will remember that in many ways I am a great fan of Einstein, and I'm enamored with his theories--even though he was wrong at times. It's easy to second-guess his contributions to the Manhattan Project, which for a time broke his life-long pacifism, and I can't judge him on that in either direction. But whatever else Albert Einstein was, he was not a man of wisdom. He was very intelligent and well-spoken, and it would be absurd for me or anyone else to diminish the contributions he has made. Unfortunately, those attributes alone do not make for wisdom. Wisdom includes the application of knowledge in one's life. An extremely smart person may make some terribly unwise decisions, and a biography of Einstein will disclose to you pretty quickly that he was a man with some serious faults.
Yes, yes, we all have serious faults, but we're not asking to be given an exemption in order to be considered prophets of an irrational mysticism. One other item that Einstein stressed frequently was that the God who legislated the laws of nature did not legislate morality and that morality was a purely human creation. Sadly, his life confirmed that his pseudo-religion did not impact his personal conduct.
Oh, how I wish he had seen the Creator's hand within creation!
Next time: Back to the Golden Ratio!
June and I were both really wound up when we came home last night after the family fireworks. Plus I really wanted to get that previous entry posted. What I thought would take about fifteen minutes took 2 hours, thanks to formatting glitches, so that set my brain twirling. (And once again, I'm back to writing my code on Notepad, and then copying and posting it to Bravenet.) We finally settled down around 4:30 am, but it was several more hours before I actually fell asleep for a few hours. June tells me that at one point during my sleep I started to sing out loud, which woke her up for a while. But she didn't remember (or recognize) the song. We've both been pretty exhausted all day today.
Please be aware of the fact that all of my previous posts in this serious are collected on my site: Phi--Let's Get it Right.
Let us move on to a second video in which the eminent theoretical physicist, Michio Kaku, declares his opinions about God.
(1) Let's ignore the gushing interviewer's comments about the "spirituality" of most cosmologists. If I were doing a one-on-one interview with someone of Dr. Kaku's stature, I, too, would treat him as the celebrity that he is. She self-edited her first question as she went along, and I think I have been able to assemble the bare bones of it without the asides and qualifications:
Q: "What is your view on life, and where is it? What are we doing when we teleport life?"
A: "Well, if I had an answer to life, I would have an inside track up there." He's pointing straight up.
Response: Kaku's answer has several dimensions (though not ten or eleven). For one thing, this answer may lead us to infer that Kaku holds to a view of God as transcendent--beyond the world and unknowable. In what follows, however, it becomes clear that Kaku's conception of God is of one who is entirely immanent, who is one with the world and its structure. Another observation I must make is that he seems to have no room for the common belief among those who believe in a transcendent God, that he has the ability to reveal himself and that he has done so. As a Christian, I can't say that I have an "inside track" to God, whatever that could mean. But I know his message and his will because he has disclosed himself to us, not in all of his infinity, of course, but in a way that becomes intelligible to us. Furthermore, what I know about him is available to anyone. When I read the Bible, I do get information about the meaning of life, not exhaustively, but enough that I'm not in the total dark about it. Obviously, non-Christians don't see it that way, and there is no need to make a big point out of that. My main objective is to help you understand what I think Kaku is trying to say, and doing so involves exposing the inadequacies of his theological concepts. If some things that he says don't make sense to you, don't blame yourself. Right here he seems to be referring to an all-transcendent, but essentially unknowable, being. But he doesn't stick to that script for very long.
(2) Prof. Kaku continues to expound his views of God. He refers to Einstein and his answer to the question of whether he believes in God. As a physicist one must, of course, be scientific in dealing with that question, and he clearly takes great pleasure in his elitist standing in such matters. The perception he seems to want to create is that he is more rational and informed than your average poorly educated believer in God. Consequently, he avers, we need to have a clear definition of what we mean by God. One cannot help but see the subtext that people who are not scientists do not live up to his supposed strictly rational analysis. There are two ways of thinking of God, he claims. (I'm treating this part of the exchange in three sections: the limit of two options, the God whom he dismisses, and the God whom he believes in.)
Response: In informal logic, we encounter a fallacy known as a false dilemma. One commits this fallacy when one presents an issue as though it had only two options, and one of them is so unacceptable that one must embrace the other one.
Your only choices are A and B. A is wonderful, while B is totally unacceptable.
"Do you want to be a benevolent pacifist and abandon all physical violence, or do you
choose to be the kind of monster who thinks that we should nuke
all the countries that we don't like and not worry about the consequences?
Hopefully, rational people understand that there are many more options between absolute pacifism and extreme war-monger-ism. That example is obviously not our topic, but simply serves as an illustration of the kind of polemic that Prof. Kaku engages in. To be sure, there may be times in our lives when one may have only two options to choose from, and one of them may be so bad that it really doesn't deserve consideration. Then there is no fallacy. The actual fallacy occurs when one imposes such a scheme on an issue while either ignoring or being ignorant of other options. My paraphrase:
There are two ways of looking at God: One comes with the clout of being held by scientific geniuses,
while the other one is childish and can be dismissed with a gentle chuckle.
If Prof. Kaku only knew what he doesn't know! There are many more ways of understanding God than a laughable cartoon version and his allegedly scientific one. What grates as much as anything is the air of superiority with which Kaku spreads his peacock feathers of ignorance to the world. He, after all, is a scientist, so he is in a far better position to understand the idea of God than the huddled masses. This is pure arrogance, particularly in light of the fact that, when we're all done we'll see that he doesn't even have any coherent understanding of God. Just to mention a few options in the conceptualization of God of which doesn't seem to be aware: personal religious theism, impersonal philosophical theism, trinitarian theism, unitarian theism, personal pantheism, impersonal pantheism, deism, panentheism of various forms (e.g., Hegel's transcendental panentheism or Whitehead's process panentheism), polytheism, henotheism, and others, not to mention combinations of some of them. I am tempted to say that before one proceeds to declare that there are only two options, one ought to have acquainted oneself with all of the options. But I cannot make that statement because undoubtedly there are many more ways of understanding God that I'm not aware of--an observation that underscores my point. But there are many models well accessible to study and learn about, and to present a picture of only two choices, one infantile and the other unintelligible, clearly demonstrates a lack of learning.
Now, one can come up with a good rebuttal to what I just presented. One could say that, regardless of the model of God one proposes, each of them clearly falls into one of two fundamental categories. But that's not what Kaku is doing. He's not giving us two classes of gods into which we can divide all the available models. He is having us choose between two specific models of God, the "God of Intervention" and the "God of Order." And that's why I must say that he is trying to entice us with a false dilemma.
(3)The first option for understanding God is as a God of intervention who answers all of our prayers, who parts the waters, who kills Philistines on our behalf as per our requests. Einstein, Prof. Kaku tells us, had a hard time believing in that kind of God.
Response: So do I. What Einstein and Kaku are dismissing is the "Santa Claus" version of God that maturing Christians should learn to grow out of. It's not the God of the Bible. Yes, the God of the Bible is a personal God who can and does act within the world he has created. But he is not a god out of, say, Greek or Hindu mythology, who changes his mind frequently in order to accommodate our demands so as to continue to be liked by us.
(4) Kaku endorses the idea of a God of order, harmony, beauty, simplicity, and elegance. A look at the physical universe bedazzles us with those qualities. God would not have needed to create the universe in that way. The God whom both Einstein and Kaku are promoting is the "God of Spinoza," a God of order and regularity.
Response: I cannot say whether either of these two men has ever actually studied Baruch "Benedict" Spinoza (163-1677) or, if I may be just a little crass, whether they understood what they were reading. To say that "Einstein read Spinoza's Ethics" does not tell us much. I shall skip the virtually mandated biography of Spinoza and give you just a bit of insight about the core of his system. He set up his philosophy in a form that resembled Euclid's Elements, drawing up axioms, theorems, and corollaries, a procedure that obviously holds a certain attraction for these scientists. Conceptually speaking, he began with René Descartes' understanding of substances, namely that substances do not have properties in themselves, but that the properties are added to an underlying substrate, which is the substance. I like to call this the "pin cushion" theory of substances. The properties are the pins that are inserted into the cushion. They can be removed or exchanged, but the underlying cushion, i.e. the true substance remains the same. Spinoza took off from the notion that substances per se have no intrinsic properties. So, we can consider one substance and declare that it is a substance, and that we know so by a direct rational intuition rather than by the properties it displays. Then we can look at a second substance and again recognize the fact that it has no intrinsic properties.
But wait! How do we know that we're not looking at the same substance for a second time? If a substance has no discernible properties that distinguish it from other substances, it is not possible for us to recognize more than one substance. There is only one substance. So then, all diversity of substances, actions, change, motion, etc. are not truly real, but are mere modalities in which the true Substance manifests itself. It's not necessary here to go through a lengthy argument demonstrating the unresolvable contradiction that runs between identifying infinite substance with finite things. The point I'm making is much simpler than that. In Spinoza's philosophy, the details of the universe, i.e. the beauty, harmony, elegance, and order of the world are swallowed up by simplicity. In fact, this simplicity is an all-absorbing one-ness. In short, as Kaku presents his picture of God, it is not the God of Spinoza. Isabelle T. mentioned on Facebook that underlying his thought may be the Buddhist concept of sunyata, the all-pervasive emptiness of all.That's a good analysis. But still, that's not the God of Einstein or Spinoza, and in Buddhism there is no such thing as a Creator who made a universe that displays beauty, order, simplicity, elegance, and harmony. Either way, Kaku's God is severely ill-defined and self-inconsistent.
(5) The attribute of simplicity is of great importance to Kaku. All the equations of physics can be written down on one sheet of paper. Better yet, the fundamental formula of string theory (Kaku's contribution, he is careful to add) is extremely short. The shorter, the more elegant. The more elegant, the more beautiful. Kaku's formula captures the essence of God, it would appear.
Response: Kaku may claim that his belief in this strange God is based on observing the order, beauty, elegance, etc. of the universe. But his conceptualization of God has yet to attain order, let alone beauty, or elegance. One feels as though there is something else that Kaku knows (and maybe Einstein did as well) that is hidden from the rest of us. One wishes for a genuine definition or description; he started out by gloating over the assumption that as a scientist he is duty-bound to start out with a clear definition of God [my paraphrase]. But such a definition is not forthcoming. I, for one would be delighted if we could start with comprehensibility, not of God, but of what Kaku is saying. Labeling his deity as the "God of Spinoza" doesn't do it because Spinoza's God is clearly not Kaku's God. Displaying his formula to the world doesn't convey anything to most of us, except that he loves simplicity, elegance, etc. But that's where we started, so that equation doesn't tell us any more about the nature and function of this God-thing. We're pretty clear on what Kaku does not believe, namely a caricature of God as a cosmic Santa Claus. The only thing we can say definitively is that we still don't know what he really means by God and that this God supposedly differs drastically from all other conceptions of God. There seems to be an inside track, after all, though the path does not appear open to most of us.
Happy Fourth of July to all relatives, friends, acquaintances, fans, colleagues and work associates of the past, present, and future. Whether you’re celebrating or not, the day is on your calendar, and I hope it is a good one for you. The question here in Smalltown USA and environs is whether the weather will clear up for fireworks tonight. Just like yesterday, it’s cold (for July), dark, cloudy, and drizzly outside. The hour-by-hour forecast has it clearing up by around 9 pm tonight, which would be the time that it started to get dark enough for our incendiary displays. We’ll see.
Late addition: The weatherman hit the proverbial nail on the head. June and I just got back from a wonderful evening with Nick & Meghan and Seth & Amber at the former’s house in the country. N&M had everything ready for hoboes for supper and strawberry shortcake for dessert, and we had some awesome fireworks. Obviously, Sunako was there, since she lives there, and S&A brought Misha, the aging whippet, and Evey, the baby Great Dane.
I have added all of the previous posts relating to this series to the Phi website. Let’s get ourselves caught up on where we are in this long and drawn-out series, beset by many a tangent. I’ve stated that a Christian, scientist or not, should see God’s hand in the beauty and order in the world that God has created, and that such beauty should extend to the world of numbers with a beauty all its own. I expressed my disappointment with those Christians who are too eager to maintain standing in the so-called community of scientists to acknowledge that even the question of whether there is scientific evidence of intelligent design in the cosmos is a legitimate one. The subtopic at this point is those scientific luminaries who appear to go in the opposite direction and seem to be finding God in the mathematical structure of the universe itself.
Michio Kaku (1947– ) is known as a successful physicist who invented string theory and some of its subsequent developments, such as super string theory. He is also an adept populariser of physics and mathematics, and, consequently, has become well known; he quite obviously enjoys talking to the world’s population at large about science and himself, not reluctant to express opinions outside of his area of expertise. As promised in the last entry, here is his YouTube video, entitled “Is God a Mathematician?”
I’m going to follow Professor Kaku fairly closely in this round. Don’t worry; I’m not going to challenge him on the validity of string theory. There are three major figures mentioned in this short talk: Newton, Einstein and Kaku. In the actual video he does not address the question whether God is a mathematician, but of what good math actually is.
Note: I have tried several ways of formatting this discussion, and I found it to work best for me if I include my responses and critiques right along with the summaries.
(1) Isaac Newton. Kaku uses Isaac Newton as an example of a time when innovations in mathematics and physics came up together.
(a) Kaku applauds Newton for asking the question: “If an apple falls, does the moon also fall?” He classifies this question as among the greatest that any member of homo sapiens has ever asked in the 6 million years since we departed from the apes.
Response: Along with all other members of our species, Kaku is entitled to his opinion, as long as we are clear that it is nothing more than his opinion. Personally, I have a problem with his evolutionary assumption, and even if I didn’t, I’m pretty sure that I would not include Newton’s concern among the top questions ever asked by humankind. But that’s a matter of perspective.
(b) Kaku’s narrative goes on: Newton’s answer to his own question was that, yes, the moon also does fall (viz. the moon is attracted to the earth by the pull of gravity). However, Newton lacked the mathematical tools to explain this idea. So, he invented calculus in order to make it possible.
Response: Is that how it happened? I don’t think so. Newton described gravity with a famous equation, oftentimes referred to as the Inverse Square Law (ISL), where G is called the gravitational constant.
The force of gravity between two objects is obtained by multiplying the mass of the two objects, dividing that number by the square of the distance between them, and multiplying the result by the gravitational constant.
I do not see any calculus here. This formula was included in Newton’s Principia Mathematica, which came out in 1687, and which did not contain any calculus. His first publication on calculus did not appear until 1693.
(c) Kaku then summarizes Newton’s conclusion: “The moon falls because of the Inverse Square Law. So does the apple.”
Response: I replayed that segment of the video several times in order to make sure I heard correctly, and the closed captions also bore me out.
“The moon falls because of the ISL.”
Scientists don’t usually say such things. The laws of science are descriptions. Often they are statistical generalizations, and frequently they appear so ironclad that there appears to be no way around them. But there is one thing that they never are, namely causal agents. The ISL describes the earth/apple and earth/moon attractions mathematically. It tells us how to calculate the gravitational force, but it does not explain the presence of a gravitational force. Both the moon and apple would still be falling, regardless of whether we had the ISL or not.
Is this just verbal nit-picking? Should I just dismiss Kaku’s phrasing as a momentary and careless slip of the tongue, something that happens to all of us from time to time? Usually I would say so, but in light of the status that Kaku eventually gives to the laws of nature a little later on, I can’t rule out that he looks at the ISL—among others laws of nature—as more than just a description. Also, as I said above, scientists are usually far too careful to use such careless phrasing, especially, I would think, in a video published to the entire world (at least the part covered by YouTube), so I can’t help but see some intentionality in Kaku’s remark.
(2) Albert Einstein. Albert Einstein is Kaku’s example of a case where a physicist was able to draw on a mathematical method that was already available prior to making an innovation in physics.
(a) Kaku tells us that Einstein came up with a new way of looking at gravity. The key to Einstein’s general theory of relativity was that he envisioned space as curved, and gravity is due to that curvature, not a force of attraction. The fact that he remains in a chair, Kaku assures us, is not because gravity pulls him, but because curved space pushes him down.
Response: I will just say that I have never heard it put that way. More commonly the explanation is that mass and energy create curvatures in space, so that my sitting in a chair causes a little protuberance in space into which I have sunk myself. Kaku seems to substitute one mysterious force for another, i.e. pushing instead of pulling, and I’m just a little confused on why he would put it that way. Still, I’m not going to argue with Kaku over matters that are purely physics. Maybe someone else can comment (nicely) on that description.
(b) According to Kaku, Einstein was also in need of a mathematical means to describe his innovation in physics. But he did not have to come up with anything new; he could simply avail himself of “differential calculus,” which already was in place. After all, he reminds us, studying calculus usually begins with the motion of falling objects, i.e. gravity.
Response: The last part of that little summary is true. Among other things, differential calculus analyzes the rate of change in the velocity of an object in motion at a particular time, e.g., the acceleration of a moving object, or even the acceleration of the acceleration. This is one part of calculus that Isaac Newton and Leibniz had already invented, and if Einstein would first have had to learn differential calculus at that point in time, he would not have been much of a physicist. (The other part of calculus is called integral calculus.)
Einstein did not need differential calculus to make his theory of space rigorous. Professor Kaku's statement implies that we begin the study of calculus with differentiation (aka “finding derivatives”), and that would most likely be true for pretty much everyone, I would think. But Einstein’s math goes far, far beyond “differential calculus.” Yes, there is calculus involved, but so are addition, subtraction, multiplication, and division. Surely all of these mathematical procedures become trivial characterizations in connection with Einstein’s work. If we were to try to cover Einstein’s math with short expressions, we might want to say that it was “tensor calculus” for the special theory of relativity and “Riemann field geometry” for the general theory. I don’t understand why Kaku trivialized Einstein’s work with the expression he used.
Could this just be a sloppy choice of words, and am I maybe nit-picking again? Once again, I doubt it. Who, given just a small amount of understanding of what is involved, would say that Einstein resorted to “differential calculus” for his general theory? This is not just careless, but misleading. One of the world’s foremost scientists must know better than to make such a mistake. My hunches: Kaku is trying to put us at ease by using a term that is not going to frighten us away. After all, we’ve all studied calculus, right? Could there be a further reasons to minimize Einstein and, thereby, establish a contrast to himself?
To get a flavor of relativity math—or maybe even to learn it—may I suggest Peter Collier, A Most Incomprehensible Thing: Notes toward a (very) gentle introduction to the mathematics of relativity (Incomprehensible Books, 2012). I must say, though, that Collier and I might just disagree on the meaning of “very gentle.” I suspect that there actually is no gentle way of learning this material.
(3) Michio Kaku. As you can see in the video, Professor Kaku is not exactly humble about his accomplishments, and I, for one, want to congratulate him on his life’s work. Even if string theory, super-string theory, M-theory, etc. are not going to hold up in the long run (and I am definitely in no position to judge that), his ideas have been significant in keeping the search for ultimate physical reality moving ahead.
(a) The holy grail, so to speak, of theoretical physics is a unified field theory, an explanation for everything. Einstein spent the majority of his life in a futile quest for it. Kaku believes that he has found it with his super-string theory. Such a theory should be able to be captured in a short equation, not more than an inch or so long. He has accomplished it, and here it is:
Response: I have no idea what this equation says, and am in no way qualified to evaluate it. It represents the vibrations of really tiny strings in the 10th and 11th dimensions, and I understand that the high degree of difficulty in understanding it is not just limited to us amateurs who like to read books about math.
(b) The math for Kaku’s string theory is analogous to that of topology, much of which was first formulated towards the end of the nineteenth century. It is an extension of algebraic geometry, reaching into higher and higher dimensions. As Kaku tells it, those who contributed to the development of topology took great delight in the fact that this was math at its best, namely, math that would never have any practical applications since it includes multiple dimensions beyond our usual three (or four if you count time).
Response: The more dimensions you give yourself to work with, the more problems you can solve. If you only had two spatial dimensions, you could have a flat depiction of the eye of a needle and some thread, but you could never thread the needle. To do that, it takes three dimensions. Topology uses that principle and develops some truly bizarre models of algebraic objects in greater spatial dimensions. I’ll leave it to Professor Kaku to judge to what extent the mathematicians delighted in the so-called lack of applicability.
(c) Along came Michio Kaku, who used 10 and 11 dimensions in order to come up with his super-string theory. Mathematicians were surprised, if not shocked and appalled.
Response: As well they should be. You’ve got to be in awe of Kaku’s work.
(d) The vibration of these strings constitute the “music of the spheres,” and the very voice of God. He, Michio Kaku, has had a glimpse of the core of the universe, where he finds God communicating to him through the equation that he, Professor Kaku, has constructed.
Response: It would appear that Michio Kaku has endowed himself with the offices of prophet and priest in the religion of super-string physics. He has departed from physics and is trying to do theology, and the result is dubious.
We need to leave the more specific content of this religion for next time.